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The change of the Weierstrass structure under one row perturbation

Itziar Baragaña

orcid.org/0000-0001-9573-9155

Departamento de Ciencia de la Computación e IA, Universidad del País Vasco, Euskal Herriko Unibertsitatea, UPV/EHU, Leioa, Spain

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Alicia Roca,

Corresponding Author

Alicia Roca

[email protected]

orcid.org/0000-0003-1926-1895

Departamento de Matemática Aplicada, IMM, Universitat Politècnica de València, Valencia, Spain

Correspondence Alicia Roca, Departamento de Matemática Aplicada, IMM, Universitat Politècnica de València, Valencia, Spain.

Email: [email protected]

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Itziar Baragaña,

Itziar Baragaña

orcid.org/0000-0001-9573-9155

Departamento de Ciencia de la Computación e IA, Universidad del País Vasco, Euskal Herriko Unibertsitatea, UPV/EHU, Leioa, Spain

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Alicia Roca,

Corresponding Author

Alicia Roca

[email protected]

orcid.org/0000-0003-1926-1895

Departamento de Matemática Aplicada, IMM, Universitat Politècnica de València, Valencia, Spain

Correspondence Alicia Roca, Departamento de Matemática Aplicada, IMM, Universitat Politècnica de València, Valencia, Spain.

Email: [email protected]

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First published: 13 November 2021

https://doi.org/10.1002/cmm4.1211

Funding information: Ministerio de Economía, Industria y Competitividad (MINECO) and Fondo Europeo de Desarrollo Regional (FEDER), Grant/Award Number: MTM2017-83624-P.

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Abstract

In this work we study the change of the structure of a regular pencil when we perform small perturbations over some of its rows and the other rows remain unaltered. We provide necessary conditions when several rows are perturbed, and prove them to be sufficient to prescribe the homogenous invariant factors or the Weyr characteristic of the resulting pencil when one row is perturbed.

1 INTRODUCTION

The additive perturbation problem of a matrix can be stated as follows: given a matrix A, analyze the structure of $A + P$ , where P is a perturbation matrix with certain properties. Different types of problems have been investigated, depending on different requirements over A and the perturbation P. Analogous problems can also be stated for a matrix pencil $A (s)$ and a perturbation pencil $P (s)$ .

Results about perturbations of square matrices where the perturbation is a matrix of bounded rank can be found in References 1-4, among others. Changes of the Weierstrass or the Kronecker structure of regular or singular pencils, respectively, under pencil perturbations of bounded rank have also been obtained (see, for instance, References 5-10 and the references therein).

Other types of problems arise when the perturbation is required to be small. Thus, changes of the Jordan structure of a square matrix under small additive perturbations were studied in References 11, 12. Small additive perturbations have also been studied for pairs of matrices,¹³ and for pencils.^{14, 15} When small additive perturbations are performed only over one or several rows, changes in the similarity invariants of a matrix and changes in the feedback invariants of a pair of matrices have also been explored.^16-19

Our target is to generalize the research of References 16-18 to matrix pencils. It is natural to pose the following problem.

Problem 1.Given a pencil $A (s) = [\begin{matrix} A_{1} (s) \\ A_{2} (s) \end{matrix}] \in ℂ {[s]}^{(r + (m - r)) \times n}$ , characterize the Kronecker structure of the pencils $A^{'} (s) = [\begin{matrix} A_{1}^{'} (s) \\ A_{2} (s) \end{matrix}] \in ℂ {[s]}^{(r + (m - r)) \times n}$ obtained from $A (s)$ under small additive perturbations over $A_{1} (s)$ .

As mentioned in Reference 16, the small perturbation problem of several rows is sort of a “crossroad” of perturbation and completion problems. On one hand, the general small perturbation problem must be taken into account. On the other hand, when perturbing one or several rows of a square matrix (see References 16-18), the problem of characterizing the invariant factors of a square matrix with some prescribed rows plays an important role. This is the problem of completion of a rectangular matrix to a square one, and was solved in Reference 20. The problem of perturbing one row in a pair of matrices (see Reference 19) involves the problem of characterizing the feedback invariants of a pair of matrices with some prescribed rows. This problem was solved in Reference 21.

For general pencils, the problem of characterizing the Kronecker structure of a matrix pencil with prescribed rows was solved in Reference 22 (see also References 23, 24).

In this article we study Problem 1 for regular pencils. We obtain necessary conditions when r rows of a regular pencil are perturbed, and solve the problem completely when $r = 1$ , hence generalizing the results of Reference 16. To solve the problem we follow the ideas of Reference 16, but we have to overcome the difficulties appearing due to the presence of infinite elementary divisors in the pencils.

The article is organized as follows. We introduce some notation and basic definitions in Section 2. Section 3 is devoted to present previous results. This section is structured in two subsections. Section 3.1 contains results on perturbation of pencils, whilst results on completion problems are included in Section 3.2. Section 4 contains the main results of this work. In Theorems 6 and 7 we obtain necessary conditions that the Weierstrass invariants must satisfy when a regular pencil is perturbed on r rows. For $r = 1$ , we prove that the necessary conditions obtained are sufficient for prescribing the homogeneous invariant factors (Theorem 8) or the Weyr characteristic (Theorem 9) of the perturbed pencil. Finally, Section 5 includes a summary of the results obtained in the article and future work.

2 NOTATION AND BASIC DEFINITIONS

We start with the introduction of some properties of integers. We call partition of a positive integer n to a finite or infinite sequence of nonnegative integers $a = (a_{1}, a_{2}, \dots)$ almost all zero, such that $a_{1} \geq a_{2} \geq \dots$ and $\sum_{i \geq 1} a_{i} = n$ . The number of components of $a$ different from zero is the length of $a$ (denoted $ℓ (a)$ ). Notice that $ℓ (a) \leq n$ . For $a = (a_{1}, \dots, a_{n})$ and $b = (b_{1}, \dots, b_{n})$ , $a$ is majorized by $b$ in the Hardy-Littlewood-Pólya sense ( $a ≺ b$ ) if $\sum_{i = 1}^{k} a_{i} \leq \sum_{i = 1}^{k} b_{i}$ for $1 \leq k \leq n - 1$ and $\sum_{i = 1}^{n} a_{i} = \sum_{i = 1}^{n} b_{i}$ . If $\sum_{i = 1}^{k} a_{i} \leq \sum_{i = 1}^{k} b_{i}$ , $1 \leq k \leq n$ , it is said that $a$ is weakly majorized by $b$ ( $a ≺ ≺ b$ ) (see Reference 25).

The conjugate partition of $a$ , $\overline{a} = (ā_{1}, ā_{2}, \dots)$ , is defined as $ā_{k} : = # {i : a_{i} \geq k}, k \geq 1$ . Given $a$ and $b$ two partitions, $a \cup b$ is the partition whose components are those of $a$ and $b$ arranged in decreasing order, and $a + b$ is the partition whose components are the sums of the corresponding components of $a$ and $b$ . The following properties are satisfied: $a ≺ b \Leftrightarrow \overline{b} ≺ \overline{a}$ and $\overline{a \cup b} = \overline{a} + \overline{b}$ .

In Reference 26 (definition 2) a generalized majorization between three finite sequences of integers, $c = (c_{1}, \dots, c_{m})$ , $a = (a_{1}, \dots, a_{s})$ , and $d = (d_{1}, \dots, d_{m + s})$ is defined and it is denoted by $d ≺^{'} (c, a)$ . When $s = 0$ , the generalized majorization reduces to $d = c$ and when $m = 0$ , to $d ≺ a$ .

Through this article, $ℂ$ denotes the field of complex numbers and $𝔽$ any arbitrary field. $𝔽 [s]$ is the ring of polynomials in the indeterminate s with coefficients in $𝔽$ and $𝔽 [s, t]$ the ring of polynomials in two variables $s, t$ with coefficients in $𝔽$ . We denote by $𝔽^{p \times q}$ , $𝔽 {[s]}^{p \times q}$ , and $𝔽 {[s, t]}^{p \times q}$ the vector spaces of $p \times q$ matrices with elements in $𝔽$ , $𝔽 [s]$ , and $𝔽 [s, t]$ , respectively. ${Gl}_{p} (𝔽)$ will be the general linear group of invertible matrices in $𝔽^{p \times p}$ .

Given a polynomial $α (s) = \sum_{i = 0}^{g} α_{i} s^{i} \in 𝔽 [s]$ , with $\deg (α) = g$ , and an integer $h \geq g$ , we will denote by ${rev}_{h} (α) (t)$ the polynomial ${rev}_{h} (α) (t) = t^{h} α (\frac{1}{t}) = t^{h - g} \sum_{i = 0}^{g} α_{i} t^{g - i} \in 𝔽 [t]$ . We have $\deg ({rev}_{h} (α)) \leq h$ . If $h = g$ , we denote ${rev}_{g} (α) = \tilde{α} (t)$ . Then, $\tilde{α} (0) \neq 0$ .

The companion matrix of a monic polynomial

α (s) = s^{n} + a_{n - 1} s^{n - 1} + \dots + a_{1} s + a_{0} \in 𝔽 [s]

, will be

C = [\begin{matrix} 0 & 1 & \dots & 0 & 0 \\ 0 & 0 & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 0 & 0 & \dots & 0 & 1 \\ - a_{0} & - a_{1} & \dots & - a_{n - 2} & - a_{n - 1} \end{matrix}] \in 𝔽^{n \times n} .

Given a polynomial matrix $A (s) \in 𝔽 {[s]}^{m \times n}$ , the degree of $A (s)$ ( $\deg (A (s))$ ) is the maximum of the degrees of its entries, and the normal rank of $A (s)$ ( $rank (A (s))$ ) is the order of the largest non identically zero minor of $A (s)$ , that is, it is the rank of $A (s)$ considered as a matrix on the field of fractions of $𝔽 [s]$ . If $\deg (A (s)) = g$ and h is an integer $h \geq g$ , then ${rev}_{h} (A) (t) = t^{h} A (\frac{1}{t}) \in 𝔽 {[t]}^{m \times n}$ .

A matrix $U (s) \in 𝔽 {[s]}^{n \times n}$ is unimodular if it is a unit in the ring $𝔽 {[s]}^{n \times n}$ , that is, $0 \neq \det (U (s)) \in 𝔽$ . Two polynomial matrices $A (s), B (s) \in 𝔽 {[s]}^{m \times n}$ are equivalent ( $A (s) \sim B (s)$ ) if there exist unimodular matrices $U (s) \in 𝔽 {[s]}^{m \times m}$ , $V (s) \in 𝔽 {[s]}^{n \times n}$ such that $B (s) = U (s) A (s) V (s)$ . If $A (s) \in 𝔽 {[s]}^{m \times n}$ and $rank (A (s)) = ρ$ , then (see e.g., Reference 27 [ch. 6]) $A (s)$ is equivalent to a unique matrix of the form $S (s) = [\begin{matrix} diag (α_{1} (s), \dots, α_{ρ} (s)) & 0 \\ 0 & 0 \end{matrix}],$ where $α_{1} (s), \dots, α_{ρ} (s)$ are monic polynomials and $α_{1} (s) | \dots | α_{ρ} (s)$ . The matrix $S (s)$ is the Smith form of $A (s)$ and the polynomials $α_{1} (s), \dots, α_{ρ} (s)$ are the invariant factors of $A (s)$ . We will take $α_{i} (s) = 1$ for $i < 1$ and $α_{i} (s) = 0$ for $i > ρ$ . For $1 \leq k \leq ρ$ , the monic greatest common divisor of the minors of $A (s)$ of order k is the determinantal divisor of $A (s)$ of order k, denoted by $D_{k} (s)$ , and $D_{k} (s) = α_{1} (s) \dots α_{k} (s)$ . The invariant factors form a complete system of invariants for the equivalence of polynomial matrices, that is, two polynomial matrices $A (s), B (s) \in 𝔽 {[s]}^{m \times n}$ are equivalent if and only if they have the same invariant factors.

A matrix pencil is a polynomial matrix $A (s) \in 𝔽 {[s]}^{m \times n}$ of degree at most one ( $A (s) = A_{0} + s A_{1}$ ). The pencil is regular if $m = n = rank (A (s))$ . Otherwise it is singular. If $rank (A (s)) = \min {m, n}$ the pencil is also called quasi-regular. The set of matrix pencils in $𝔽 {[s]}^{m \times n}$ is denoted by $𝒫_{m \times n} (𝔽)$ .

Two matrix pencils $A (s) = A_{0} + s A_{1}, B (s) = B_{0} + s B_{1} \in 𝒫_{m \times n} (𝔽)$ are strictly equivalent ( $A (s) \overset{s . e .}{\sim} B (s)$ ) if there exist invertible matrices $P \in {Gl}_{m} (𝔽)$ , $Q \in {Gl}_{n} (𝔽)$ such that $B (s) = P A (s) Q$ .

Given the pencil

A (s) = A_{0} + s A_{1} \in 𝒫_{m \times n} (𝔽)

rank A (s) = ρ

, a complete system of invariants for the strict equivalence is formed by a chain of homogeneous polynomials

ϕ_{1} (s, t) | \dots | ϕ_{ρ} (s, t), ϕ_{i} (s, t) \in 𝔽 [s, t], 1 \leq i \leq ρ

, monic with respect to s, called the homogeneous invariant factors and two finite partitions of nonnegative integers,

c_{1} \geq \dots \geq c_{n - ρ}

and

u_{1} \geq \dots \geq u_{m - ρ}

, called the column and row minimal indices of the pencil, respectively. In turn, the homogeneous invariant factors are determined by the invariant factors

α_{1} (s) | \dots | α_{ρ} (s)

and a chain of polynomials

t^{k_{1}} | \dots | t^{k_{ρ}}

𝔽 [t]

, called the infinite elementary divisors (see References 28 [ch. 2] or 27 [ch. 12]). In fact, we can write

\begin{align} ϕ_{i} (s, t) = t^{k_{i}} t^{\deg (α_{i})} α_{i} (\frac{s}{t}), 1 \leq i \leq ρ . \end{align}

()

Observe that

α_{i} (s) = ϕ_{i} (s, 1)

1 \leq i \leq ρ

. If

{\overline{α}}_{1} (t) | \dots | {\overline{α}}_{ρ} (t)

are the invariant factors of the pencil

{rev}_{1} (A) (t) = t A_{0} + A_{1} \in 𝔽 {[t]}^{m \times n}

, then for some

0 \neq l_{i} \in 𝔽

\begin{align} l_{i} {\overline{α}}_{i} (t) = ϕ_{i} (1, t) = t^{k_{i}} {\tilde{α}}_{i} (t), 1 \leq i \leq ρ . \end{align}

()

(Recall that

{\tilde{α}}_{i} (t) = {rev}_{g_{i}} (α_{i})

, where

g_{i} = \deg (α_{i})

). If

A (s) \in 𝒫_{m \times n} (𝔽)

and

rank (A (s)) = m

(

rank (A (s)) = n

), then

A (s)

does not have row (column) minimal indices. As a consequence, the invariants for the strict equivalence of regular matrix pencils are reduced to the homogeneous invariant factors.

A canonical form for the strict equivalence of matrix pencils is the Kronecker canonical form. Let

A (s) \in 𝒫_{m \times n} (𝔽)

be a pencil of rank

A (s) = ρ

, with invariant factors

1 = α_{1} (s) = \dots = α_{ρ - x} (s) \neq α_{ρ - x + 1} (s) | \dots | α_{ρ} (s)

, where

0 \leq x \leq ρ

and

\deg (α_{ρ - x + i}) = g_{i} > 0

1 \leq i \leq x

, infinite elementary divisors

1 = t^{k_{1}} = \dots = t^{k_{ρ - y}} \neq t^{k_{ρ - y + 1}} | \dots | t^{k_{ρ}}

, where

0 \leq y \leq ρ

, column minimal indices

c_{1} \geq \dots \geq c_{r} > 0 = c_{r + 1} = \dots = c_{n - ρ}

, where

0 \leq r \leq n - ρ

, and row minimal indices

u_{1} \geq \dots \geq u_{z} > 0 = u_{z + 1} = \dots = u_{m - ρ}

, where

0 \leq z \leq m - ρ

. Then the Kronecker canonical form of

A (s)

[\begin{matrix} L (s) & 0 & 0 & 0 & 0 \\ 0 & R (s) & 0 & 0 & 0 \\ 0 & 0 & C (s) & 0 & 0 \\ 0 & 0 & 0 & N (s) & 0 \\ 0 & 0 & 0 & 0 & 0 \end{matrix}] \in 𝒫_{m \times n} (𝔽),

where

L (s) = diag (L_{c_{1}} (s), \dots, L_{c_{r}} (s))

R (s) = diag (R_{u_{1}} (s), \dots, R_{u_{z}} (s))

C (s) = diag (s I_{g_{1}} - C_{1}, \dots, s I_{g_{x}} - C_{x})

N (s) = diag (N_{k_{ρ - y + 1}} (s), \dots, N_{k_{ρ}} (s))

, with

C_{i}

the companion matrix of

α_{ρ - x + i}

1 \leq i \leq x

\begin{align} L_{k} (s) = [\begin{matrix} s & - 1 \\ ⋱ & ⋱ \\ s & - 1 \end{matrix}] \in 𝒫_{k \times (k + 1)} (𝔽), R_{k} (s) = L_{k}^{T} (s) \in 𝒫_{(k + 1) \times k} (𝔽), N_{k} (s) = [\begin{matrix} 1 & - s \\ ⋱ & ⋱ \\ ⋱ & - s \\ 1 \end{matrix}] \in 𝒫_{k \times k} (𝔽), \end{align}

()

understanding that the non specified components are zero. For details see References 28 (ch. 2) or 27 (ch. 12) for infinite fields, and Reference 29 (ch. 2) for arbitrary fields. From the Kronecker canonical form of the pencil

A (s)

it is easy to see that

\sum_{i = 1}^{ρ} \deg (ϕ_{i}) + \sum_{i = 1}^{n - ρ} c_{i} + \sum_{i = 1}^{m - ρ} u_{i} = ρ

. If

A (s) \in 𝒫_{p \times p} (𝔽)

is a regular pencil, then the Kronecker canonical form of

A (s)

is reduced to

diag (C (s), N (s))

and is known as the Weierstrass canonical form.

Assume now that $A (s) = A_{0} + s A_{1} \in 𝒫_{m \times n} (ℂ)$ is a complex matrix pencil. Let $\overline{ℂ} = ℂ \cup {\infty}$ . The spectrum of $A (s)$ is $Λ (A (s)) = {λ \in \overline{ℂ} : rank (A (λ)) < rank (A (s))},$ where we agree that $A (\infty) = A_{1}$ . The elements $λ \in Λ (A (s))$ are the eigenvalues of $A (s)$ .

We can factorize

\begin{align} α_{ρ - i + 1} (s) = \prod_{λ \in Λ (A (s)) ∖ {\infty}} {(s - λ)}^{n_{i} (λ, A (s))}, 1 \leq i \leq ρ, \\ ϕ_{ρ - i + 1} (s, t) = t^{n_{i} (\infty, A (s))} \prod_{λ \in Λ (A (s)) ∖ {\infty}} {(s - λ t)}^{n_{i} (λ, A (s))}, 1 \leq i \leq ρ . \end{align}

The integers

n_{1} (λ, A (s)) \geq \dots \geq n_{ρ} (λ, A (s))

are the partial multiplicities of

λ

A (s)

n (λ, A (s)) = (n_{1} (λ, A (s)), \dots, n_{ρ} (λ, A (s)))

is the partition of

λ

in the Segre characteristic of

A (s)

and its conjugate partition

w (λ, A (s)) = \overline{n (λ, A (s))} = (w_{1} (λ, A (s)), \dots, w_{ρ} (λ, A (s)))

is the partition of

λ

in the Weyr characteristic of

A (s)

For $λ \in \overline{ℂ} ∖ Λ (A (s))$ we take $n (λ, A (s)) = w (λ, A (s)) = 0$ . We agree that $n_{i} (λ, A (s)) = + \infty$ for $i < 1$ and $n_{i} (λ, A (s)) = 0$ for $i > ρ$ , for $λ \in \overline{ℂ}$ . We also agree that $ϕ_{i} (s, t) = 1$ for $i < 1$ and $ϕ_{i} (s, t) = 0$ for $i > ρ$ . Observe that, in (1), $k_{i} = n_{ρ - i + 1} (\infty, A (s))$ , $1 \leq i \leq ρ$ , and from (2) we conclude that $n (\infty, A (s)) = n (0, {rev}_{1} (A) (t))$ , thus $w (\infty, A (s)) = w (0, {rev}_{1} (A) (t))$ .

We will use the $ℓ_{1}$ norm in the vector space of polynomials of degree less than or equal to n. Given a polynomial $α (s) = a_{n} s^{n} + a_{n - 1} s^{n - 1} + \dots + a_{1} s + a_{0} \in ℂ [s]$ , $‖ α (s) ‖ = \sum_{i = 1}^{n} | a_{i} |$ . Notice that if $α (s), β (s) \in ℂ [s]$ and h is an integer such that $\deg (α) \leq h$ , $\deg (β) \leq h$ , then $‖ α (s) - β (s) ‖ = ‖ {rev}_{h} (α) (t) - {rev}_{h} (β) (t) ‖$ .

Given a matrix $M (s) = (m_{i, j} (s)) \in ℂ {[s]}^{m \times n}$ , $‖ M (s) ‖ = \sum_{i = 1}^{m} \sum_{j = 1}^{n} ‖ m_{i, j} (s) ‖$ . If $M (s) \in ℂ {[s]}^{m \times n}, N (s) \in ℂ {[s]}^{n \times p}$ , then $‖ M (s) N (s) ‖ \leq ‖ M (s) ‖ ‖ N (s) ‖ .$

Given a real number $η > 0$ and $λ \in ℂ$ , $B (λ, η) = {z \in ℂ : | z - λ | < η}$ denotes the open ball centered at $λ$ and radius $η$ . For $λ = \infty$ , $B (\infty, η) = {z \in ℂ : | z | > η^{- 1}} \cup {\infty} .$

For a given pencil $A (s) \in 𝒫_{m \times n} (ℂ)$ , we define the $η$ -neighborhood of the spectrum of $A (s)$ as $𝒱_{η} (A (s)) {= ⋃}_{λ \in Λ (A (s))} B (λ, η),$ whenever the balls $B (λ, η)$ are pairwise disjoint.

3 PREVIOUS RESULTS

In this section, we present some preliminary results. We have grouped them in two subsections.

3.1 Perturbation results

First of all, we show that in Problem 1 we can assume that $A_{2} (s)$ is in Kronecker canonical form. The proof follows the scheme of that of Reference 16 (lemma 3.2).

Lemma 1.Let $A (s) = [\begin{matrix} A_{1} (s) \\ A_{2} (s) \end{matrix}] \in 𝒫_{(r + (m - r)) \times n} (ℂ)$ and $B (s) \in 𝒫_{m \times n} (ℂ)$ be matrix pencils.

Let $Ā_{2} (s) = P A_{2} (s) Q$ and $Ā_{1} (s) = A_{1} (s) Q$ with $P \in {Gl}_{m - r} (ℂ)$ , $Q \in {Gl}_{n} (ℂ)$ . Let $Ā (s) = [\begin{matrix} Ā_{1} (s) \\ Ā_{2} (s) \end{matrix}]$ . The following propositions are equivalent:

(i)
For every $ϵ > 0$ , there exists a pencil $A^{'} (s) = [\begin{matrix} A_{1}^{'} (s) \\ A_{2} (s) \end{matrix}] \in 𝒫_{(r + (m - r)) \times n} (ℂ)$ such that $‖ A^{'} (s) - A (s) ‖ < ϵ$ and $A^{'} (s) \overset{s . e .}{\sim} B (s)$ .
(ii)
For every $ϵ^{'} > 0$ , there exists a pencil $Ā^{'} (s) = [\begin{matrix} Ā_{1}^{'} (s) \\ Ā_{2} (s) \end{matrix}] \in 𝒫_{(r + (m - r)) \times n} (ℂ)$ such that $‖ Ā^{'} (s) - Ā (s) ‖ < ϵ^{'}$ and $Ā^{'} (s) \overset{s . e .}{\sim} B (s)$ .

Proof.We have $diag (I_{r}, P) A (s) Q = Ā (s)$ .

$\underline{(i) \Rightarrow (i i) :}$ Let $ϵ^{'} > 0$ . From (i) we know that given $ϵ > 0$ there exists a pencil $A^{'} (s) = [\begin{matrix} A_{1}^{'} (s) \\ A_{2} (s) \end{matrix}] \in 𝒫_{(r + (m - r)) \times n} (ℂ)$ such that $A^{'} (s) \overset{s . e .}{\sim} B (s)$ and $‖ A^{'} (s) - A (s) ‖ < ϵ$ . Let $Ā^{'} (s) = diag (I_{r}, P) A^{'} (s) Q$ . Then, $Ā^{'} (s) \overset{s . e .}{\sim} A^{'} (s) \overset{s . e .}{\sim} B (s)$ and

‖ Ā^{'} (s) - Ā (s) ‖ = ‖ diag (I_{r}, P) (A^{'} (s) - A (s)) Q ‖ \leq ‖ diag (I_{r}, P) ‖ ‖ A^{'} (s) - A (s) ‖ ‖ Q ‖ < ϵ ‖ diag (I_{r}, P) ‖ ‖ Q ‖ .

Taking

ϵ = \frac{ϵ^{'}}{‖ diag (I_{r}, P) ‖ ‖ Q ‖}

, the result follows.

$\underline{(i i) \Rightarrow (i) :}$ The proof is analogous.

The following lemma can also be found in Reference 16 (lemma 2.1).

Lemma 2. ([30], theorem VI.1.2)Let $α (s) \in ℂ [s]$ be a polynomial of degree g, $α (s) = \sum_{i = 0}^{g} a_{i} s^{i} = a_{g} (s - μ_{1}) \dots (s - μ_{g}) .$

(a)
For every $ϵ > 0$ , there exists $δ > 0$ such that if $α^{'} (s)$ is a polynomial of degree at most g and $‖ α^{'} (s) - α (s) ‖ < δ$ , then the roots of $α^{'} (s)$ are in $⋃_{i = 1}^{g} B (μ_{i}, ϵ) .$
(b)
Reciprocally, given $ϵ > 0$ there exists $δ > 0$ such that if $μ_{i}^{'} \in B (μ_{i}, δ)$ , $1 \leq i \leq g$ , and $α^{'} (s) = a_{g} (s - μ_{1}^{'}) \dots (s - μ_{g}^{'})$ , then $‖ α^{'} (s) - α (s) ‖ < ϵ$ .

In the next theorem, necessary conditions are given for perturbations of quasi-regular pencils.

Theorem 1. ([14] ch. 2, theorem 2.6, [15] theorem 4.2, particular case)Let $A (s) \in 𝒫_{m \times n} (ℂ)$ be a pencil such that $rank (A (s)) = \min {m, n}$ . Let the partition $r$ (the partition $s$ ) be the conjugate partition of that of the column (row) minimal indices of $A (s)$ .

Let $𝒱_{η} (A (s))$ be an $η$ -neighborhood of the spectrum of $A (s)$ . There exists $δ > 0$ such that if $‖ A^{'} (s) - A (s) ‖ < δ$ , then $rank (A^{'} (s)) = rank (A (s))$ and

(i)
If the partition $r^{'}$ (the partition $s^{'}$ ) is the conjugate partition of that of the column (row) minimal indices of $A^{'} (s)$ , then $r ≺ ≺ r^{'}$ ( $s ≺ ≺ s^{'}$ ).
(ii)

$\begin{align} Λ (A^{'} (s)) \subseteq 𝒱_{η} (A (s)), \end{align}$ ()
(iii)
$⋃_{μ \in B (λ, η)} w (μ, A^{'} (s)) ≺ ≺ w (λ, A (s))$ , for every $λ \in Λ (A (s)) .$

Remark 1.If $rank (A (s)) = m = n$ , that is, if $A (s)$ is regular, then $A^{'} (s)$ is also regular, condition (i) disappears and condition (iii) becomes

\begin{align} ⋃_{μ \in B (λ, η)} w (μ, A^{'} (s)) ≺ w (λ, A (s)), for every λ \in Λ (A (s)) . \end{align}

()

The following results on perturbation of matrix pencils are stated for more general pencils in the corresponding references; we present here the particular cases for regular pencils.

Theorem 2. ([15], theorem 5.1, particular case)Let $A (s) \in 𝒫_{n \times n} (ℂ)$ be a regular pencil, and let $𝒱_{η} (A (s))$ be an $η$ -neighborhood of the spectrum of $A (s)$ . For every $λ \in Λ (A (s))$ , let $t_{λ}$ be a given integer $t_{λ} \geq 0$ and let $w^{(λ, j)} = (w_{1}^{(λ, j)}, \dots)$ be given partitions, $1 \leq j \leq t_{λ}$ . For every $ϵ > 0$ , there exists a pencil $A^{'} (s) \in 𝒫_{n \times n} (ℂ)$ such that $‖ A^{'} (s) - A (s) ‖ < ϵ$ , its spectrum satisfies (4), and

\begin{align} A^{'} (s) has just t_{λ} eigenvalues μ_{λ, 1}, \dots, μ_{λ, t_{λ}} in B (λ, η) with w (μ_{λ, j}, A^{'} (s)) = w^{(λ, j)}, 1 \leq j \leq t_{λ}, for every λ \in Λ (A (s)), \end{align}

()

if and only if

\begin{align} ⋃_{j = 1}^{t_{λ}} w^{(λ, j)} ≺ w (λ, A (s)), for every λ \in Λ (A (s)) . \end{align}

()

Theorem 3. ([14], ch. 2, theorem 3.1, particular case)Let $A (s), B (s) \in 𝒫_{n \times n} (ℂ)$ be regular pencils. For every $ϵ > 0$ , there exists a pencil $A^{'} (s) \in 𝒫_{n \times n} (ℂ)$ such that $‖ A^{'} (s) - A (s) ‖ < ϵ$ and $A^{'} (s) \overset{s . e .}{\sim} B (s)$ if and only if

\begin{align} w (λ, B (s)) ≺ w (λ, A (s)), for every λ \in \overline{ℂ} . \end{align}

()

Corollary 1.Let $A (s), B (s) \in 𝒫_{n \times n} (ℂ)$ be regular pencils with homogeneous invariant factors $ψ_{1} (s, t) | \dots | ψ_{n} (s, t)$ and $ψ_{1}^{'} (s, t) | \dots | ψ_{n}^{'} (s, t)$ , respectively. For every $ϵ > 0$ , there exists a pencil $A^{'} (s) \in 𝒫_{n \times n} (ℂ)$ such that $‖ A^{'} (s) - A (s) ‖ < ϵ$ and $A^{'} (s) \overset{s . e .}{\sim} B (s)$ if and only if

\begin{align} \prod_{j = 1}^{i} ψ_{j}^{'} (s, t) | \prod_{j = 1}^{i} ψ_{j} (s, t), 1 \leq i \leq n . \end{align}

()

Proof.Taking into account that $\deg (\prod_{j = 1}^{n} ψ_{j} (s, t)) = \deg (\prod_{j = 1}^{n} ψ_{j}^{'} (s, t)) = n$ , from (9), for $i = n$ we obtain

\begin{align} \prod_{j = 1}^{n} ψ_{j}^{'} (s, t) = \prod_{j = 1}^{n} ψ_{j} (s, t) . \end{align}

()

By Theorem 3, we must prove that (9) and (10) are equivalent to (8). It is easy to see that (9) and (10) are equivalent to

\begin{align} \prod_{j = 1}^{i} ψ_{n - j + 1} (s, t) | \prod_{j = 1}^{i} ψ_{n - j + 1}^{'} (s, t), 1 \leq i \leq n, \prod_{j = 1}^{n} ψ_{n - j + 1} (s, t) = \prod_{j = 1}^{n} ψ_{n - j + 1}^{'} (s, t) . \end{align}

()

ψ_{n - i + 1} (s, t) = t^{n_{i} (\infty, A (s))} \prod_{λ \in Λ (A (s)) ∖ {\infty}} {(s - λ t)}^{n_{i} (λ, A (s))}, ψ_{n - i + 1}^{'} (s) = t^{n_{i} (\infty, B (s))} \prod_{λ \in Λ (B (s)) ∖ {\infty}} {(s - λ t)}^{n_{i} (λ, B (s))}, 1 \leq i \leq n,

condition (11) is equivalent to

\begin{align} (n (λ, A (s)) ≺ (n (λ, B (s)), for every λ \in \overline{ℂ}, \end{align}

()

which is equivalent to (8).

Remark 2.Following Reference 18 we denote condition (9) by $(ψ_{1}^{'} (s, t), \dots, ψ_{n}^{'} (s, t)) ≺ ≺ (ψ_{1} (s, t), \dots, ψ_{n} (s, t))$ , and conditions (9) and (10) by

\begin{align} (ψ_{1}^{'} (s, t), \dots, ψ_{n}^{'} (s, t)) ≺ (ψ_{1} (s, t), \dots, ψ_{n} (s, t)) . \end{align}

()

3.2 Completion results

The following theorem contains a solution of a polynomial matrix completion problem.

Theorem 4. ([3, 31])Let $A (s) \in 𝔽 {[s]}^{(m - r) \times (n - s)}$ , $B (s) \in 𝔽 {[s]}^{m \times n}$ be polynomial matrices matrices with invariant factors $α_{1} (s) | \dots | α_{ρ} (s)$ and $β_{1} (s) | \dots | β_{\overline{ρ}} (s)$ , respectively, where $ρ = rank (A (s))$ and $\overline{ρ} = rank (B (s))$ . Then, there exist $X (s) \in 𝔽 {[s]}^{r \times (n - s)}$ , $Y (s) \in 𝔽 {[s]}^{r \times s}$ , $Z (s) \in 𝔽 {[s]}^{(m - r) \times s}$ such that $B (s) \sim [\begin{matrix} X (s) & Y (s) \\ A (s) & Z (s) \end{matrix}]$ if and only if

β_{i} (s) | α_{i} (s) | β_{i + r + s} (s), 1 \leq i \leq ρ .

As mentioned, the problem of row completion of matrix pencils was solved in References 22, 23. We state here the version of Reference 24 for pencils without row minimal indices.

Theorem 5. ([24], theorem 2, particular case)Let $A_{2} (s) \in 𝒫_{(n - r) \times (n + m)} (𝔽)$ be a matrix pencil of $rank (A_{2} (s)) = n - r$ , with homogeneous invariant factors $ϕ_{1} (s, t) | \dots | ϕ_{n - r} (s, t)$ and column minimal indices $c_{1} \geq \dots \geq c_{r + m}$ . Let $A (s) \in 𝒫_{n \times (n + m)} (𝔽)$ be a matrix pencil of $rank (A (s)) = n$ with homogeneous invariant factors $ψ_{1} (s, t) | \dots | ψ_{n} (s, t)$ and column minimal indices $d_{1} \geq \dots \geq d_{m}$ . Let $c = (c_{1}, \dots, c_{r + m})$ , $d = (d_{1}, \dots, d_{m})$ and $a = (a_{1}, \dots, a_{r})$ , where

a_{i} = \deg (τ_{r - i + 1} (s, t)) - \deg (τ_{r - i} (s, t)) - 1, 1 \leq i \leq r,

with

τ_{j} (s, t) = \prod_{i = 1}^{n - r + j} lcm (ϕ_{i - j} (s, t), ψ_{i} (s, t)),

0 \leq j \leq r .

There exists a matrix pencil $A_{1} (s) \in 𝒫_{r \times (n + m)} (𝔽)$ such that $[\begin{matrix} A_{1} (s) \\ A_{2} (s) \end{matrix}] \overset{s . e .}{\sim} A (s)$ if and only if

\begin{align} ψ_{i} (s, t) | ϕ_{i} (s, t) | ψ_{i + r} (s, t), 1 \leq i \leq n - r, \end{align}

()

and

\begin{align} c ≺^{'} (d, a) . \end{align}

()

Remark 3.If $m = 0$ , then $A (s)$ is regular. As it has no column minimal indices, condition (15) becomes

c ≺ a .

4 MAIN RESULTS

In this section we study Problem 1 when $m = n$ and the pencil $A (s) = [\begin{matrix} A_{1} (s) \\ A_{2} (s) \end{matrix}] \in 𝒫_{(r + (n - r)) \times n} (ℂ)$ is regular, hence $A_{2} (s) \in 𝒫_{(n - r) \times n} (ℂ)$ is quasi-regular with $rank (A_{2} (s)) = n - r$ . We denote the homogeneous invariant factors and column minimal indices of $A_{2} (s)$ by $ϕ_{1} (s, t) | \dots | ϕ_{n - r} (s, t)$ and $c_{1} \geq \dots \geq c_{r}$ , respectively, and the homogeneous invariant factors of $A (s)$ are $ψ_{1} (s, t) | \dots | ψ_{n} (s, t)$ .

In the following results we give some necessary conditions a pencil must satisfy when obtained after perturbation of $A_{1} (s)$ . The next theorem is straightforward from Theorems 1, 5 and Remarks 1 and 3.

Theorem 6.Let $𝒱_{η} (A (s))$ be an $η$ -neighborhood of the spectrum of $A (s)$ . There exists $ϵ > 0$ such that if $A^{'} (s) = [\begin{matrix} A_{1}^{'} (s) \\ A_{2} (s) \end{matrix}] \in 𝒫_{(r + (n - r)) \times n} (ℂ)$ has $ψ_{1}^{'} (s, t) | \dots | ψ_{n}^{'} (s, t)$ as homogeneous invariant factors and $‖ A^{'} (s) - A (s) ‖ < ϵ$ , then the spectrum of $A^{'} (s)$ satisfies (4), (5), and

\begin{align} ψ_{i}^{'} (s, t) | ϕ_{i} (s, t) | ψ_{i + r}^{'} (s, t), 1 \leq i \leq n - r, \end{align}

()

and

\begin{align} c ≺ a^{'}, \end{align}

()

where

c = (c_{1}, \dots, c_{r})

and

a^{'} = (a_{1}^{'}, \dots, a_{r}^{'})

, with

a_{i}^{'} = \deg (τ_{r - i + 1}^{'} (s, t)) - \deg (τ_{r - i}^{'} (s, t)) - 1, 1 \leq i \leq r,

and

τ_{j}^{'} (s, t) = \prod_{i = 1}^{n - r + j} lcm (ϕ_{i - j} (s, t), ψ_{i}^{'} (s, t))

0 \leq j \leq r

From Corollary 1 (see Remark 2), Theorem 5 and Remark 3, we obtain

Theorem 7.Let $ψ_{1}^{'} (s, t) | \dots | ψ_{n}^{'} (s, t)$ be homogeneous polynomials, monic with respect to s. If for every $ϵ > 0$ , there exists a pencil $A^{'} (s) = [\begin{matrix} A_{1}^{'} (s) \\ A_{2} (s) \end{matrix}] \in 𝒫_{(r + (n - r)) \times n} (ℂ)$ with $ψ_{1}^{'} (s, t) | \dots | ψ_{n}^{'} (s, t)$ as homogeneous invariant factors and $‖ A^{'} (s) - A (s) ‖ < ϵ$ , then (16), (17), and (13) hold.

In the rest of the section we will assume that $r = 1$ , that is, $A (s) = [\begin{matrix} a (s) \\ A_{2} (s) \end{matrix}] \in 𝒫_{(1 + (n - 1)) \times n} (ℂ)$ , and $ψ_{1}^{'} (s, t) | \dots | ψ_{n}^{'} (s, t)$ will be prescribed homogeneous polynomials, monic with respect to s.

Since rank

A_{2} (s) = n - 1

, the pencil

A_{2} (s)

has only one column minimal index,

c_{1} \geq 0

. Let us assume that

A_{2} (s)

has

p \geq 0

nontrivial invariant factors

1 \neq α_{1} (s) | \dots | α_{p} (s)

α_{i} (s) = s^{g_{i}} + \sum_{j = 0}^{g_{i} - 1} a_{i, j} s^{j}

g_{i} > 0

1 \leq i \leq p

, and

q \geq 0

nontrivial infinite elementary divisors

t^{μ_{1}} | \dots | t^{μ_{q}}

μ_{i} > 0

1 \leq i \leq q

. Notice that if

{\hat{α}}_{1} (s) | \dots | {\hat{α}}_{n - 1} (s)

are the invariant factors of

A_{2} (s)

, then

{\hat{α}}_{i} (s) = 1

for

1 \leq i \leq n - 1 - p

and

{\hat{α}}_{n - 1 - p + i} (s) = α_{i} (s)

for

1 \leq i \leq p

. Analogously, if

t^{{\hat{μ}}_{1}} | \dots | t^{{\hat{μ}}_{n - 1}}

are the infinite elementary divisors of

A_{2} (s)

, then

{\hat{μ}}_{i} = 0

for

1 \leq i \leq n - 1 - q

and

{\hat{μ}}_{n - 1 - q + i} = μ_{i}

for

1 \leq i \leq q

. The next identities are satisfied

c_{1} + \sum_{i = 1}^{n - 1} \deg (ϕ_{i}) = c_{1} + \sum_{i = 1}^{p} g_{i} + \sum_{i = 1}^{q} μ_{i} = n - 1 .

In the case that

\sum_{i = 1}^{n} \deg (ψ_{i}^{'}) = n

and (16) holds for

r = 1

, that is,

\begin{align} ψ_{i}^{'} (s, t) | ϕ_{i} (s, t) | ψ_{i + 1}^{'} (s, t), 1 \leq i \leq n - 1, \end{align}

()

we have

τ_{1}^{'} (s, t) = \prod_{i = 1}^{n} lcm (ϕ_{i - 1} (s, t), ψ_{i}^{'} (s, t)) = \prod_{i = 1}^{n} ψ_{i}^{'} (s, t)

and

τ_{0}^{'} (s, t) = \prod_{i = 1}^{n - 1} lcm (ϕ_{i} (s, t), ψ_{i}^{'} (s, t)) = \prod_{i = 1}^{n - 1} ϕ_{i} (s, t)

, hence

a_{1}^{'} = \deg (τ_{1}^{'}) - \deg (τ_{0}^{'}) - 1 = n - (n - 1 - c_{1}) - 1 = c_{1} .

Therefore, for

r = 1

, (13) and (18) imply (17).

Our aim is to prove that conditions (13) and (18) are sufficient to guarantee that in every neighborhood of $A (s)$ there exists a pencil $A^{'} (s) \in 𝒫_{n \times n} (ℂ)$ with $ψ_{1}^{'} (s, t) | \dots | ψ_{n}^{'} (s, t)$ as homogeneous invariant factors.

By Lemma 1, we can assume that

A_{2} (s)

is in Kronecker canonical form. There are two cases to consider depending on the value of

c_{1}

\begin{align} c_{1} > 0, & A_{2} (s) = diag (L_{c_{1}} (s), C (s), N (s)) \in 𝒫_{(n - 1) \times n} (ℂ), \\ c_{1} = 0, & A_{2} (s) = [\begin{matrix} O & diag (C (s), N (s)) \end{matrix}] \in 𝒫_{(n - 1) \times n} (ℂ), \end{align}

()

where

C (s) = diag (s I_{g_{1}} - C_{1}, \dots, s I_{g_{p}} - C_{p})

N (s) = diag (N_{μ_{1}} (s), \dots, N_{μ_{q}} (s))

, with

C_{i}

the companion matrix of

α_{i} (s)

1 \leq i \leq p

, and

L_{c_{1}} (s)

and

N_{μ_{i}} (s)

1 \leq i \leq q

, are defined in (3).

The proof of the following lemma is analogous to that of Lemma 3.4 of Reference 16.

Lemma 3.Let $A (s) = A_{0} + s A_{1} = [\begin{matrix} a (s) \\ A_{2} (s) \end{matrix}] \in 𝒫_{(1 + (n - 1)) \times n} (ℂ)$ be a regular pencil, with $A_{2} (s)$ as (19).

We partition $a (s)$ according to the blocks of $A_{2} (s)$ ,

\begin{align} a (s) & = [\begin{matrix} a^{L} (s) & a^{C} (s) & a^{N} (s) \end{matrix}] \\ = [\begin{array}{c} a_{1}^{L} (s) . . . a_{c_{1} + 1}^{L} (s) ||a_{1, 1}^{C} (s) . . . a_{1, g_{1}}^{C} (s) | . . . | a_{p, 1}^{C} (s) . . . a_{p, g_{p}}^{C} (s)|| a_{1, 1}^{N} (s) . . . a_{1, μ_{1}}^{N} (s) | . . . | \end{array} a_{q, 1}^{N} (s) . . . a_{q, μ_{q}}^{N} (s)] . \end{align}

Then, there exist unimodular matrices

U (s), V (s) \in ℂ {[s]}^{n \times n}

and

Ū (t), \overline{V} (t) \in ℂ {[t]}^{n \times n}

such that

\begin{align} U (s) A (s) V (s) = diag (I_{n - 1 - p}, M (s)), Ū (t) {rev}_{1} (A) (t) \overline{V} (t) = diag (I_{n - 1 - p - q}, \overline{M} (t)), \end{align}

()

with

\begin{align} M (s) & = [\begin{array}{cccc} γ^{L} (s) & γ_{1}^{C} (s) & \dots & γ_{p}^{C} (s) \\ 0 & α_{1} (s) & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & α_{p} (s) \end{array}], \\ \overline{M} (t) & = [\begin{array}{ccccccc} {rev}_{c_{1} + 1} (γ^{L}) (t) & {rev}_{g_{1}} (γ_{1}^{C}) (t) & \dots & {rev}_{g_{p}} (γ_{p}^{C}) (t) & {rev}_{μ_{1}} (γ_{1}^{N}) (t) & \dots & {rev}_{μ_{q}} (γ_{q}^{N}) (t) \\ 0 & {\tilde{α}}_{1} (t) & \dots & 0 & 0 & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & {\tilde{α}}_{p} (t) & 0 & \dots & 0 \\ 0 & 0 & \dots & 0 & t^{μ_{1}} & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & 0 & 0 & \dots & t^{μ_{q}} \end{array}], \end{align}

()

where

\begin{align} γ^{L} (s) = \sum_{j = 1}^{c_{1} + 1} s^{j - 1} a_{j}^{L} (s), γ_{i}^{C} (s) = \sum_{j = 1}^{g_{i}} s^{j - 1} a_{i, j}^{C} (s), 1 \leq i \leq p, γ_{i}^{N} (s) = \sum_{j = 1}^{μ_{i}} s^{μ_{i} - j} a_{i, j}^{N} (s), 1 \leq i \leq q . \end{align}

()

Remark 4.

In Lemma 3,
$\begin{align} \deg (γ^{L}) \leq c_{1} + 1, \deg (γ_{i}^{C}) \leq g_{i}, 1 \leq i \leq p, \deg (γ_{i}^{N}) \leq μ_{i}, 1 \leq i \leq q . \end{align}$ ()
The pencil $A (s)$ (the pencil ${rev}_{1} (A) (t)$ ) has the same nontrivial invariant factors as the polynomial matrix $M (s)$ (the polynomial matrix $\overline{M} (t)$ ).
By Theorem 5, the pencil $A (s)$ has at most $p + 1$ nontrivial invariant factors $β_{1} (s) | \dots | β_{p + 1} (s)$ and $q + 1$ nontrivial infinite elementary divisors $t^{η_{1}} | \dots | t^{η_{q + 1}}$ satisfying
$\begin{align} β_{i} (s) | α_{i} (s) | β_{i + 1} (s), 1 \leq i \leq p, η_{i} \leq μ_{i} \leq η_{i + 1}, 1 \leq i \leq q . \end{align}$ ()
We have $α_{1} (s) \dots α_{p} (s) γ^{L} (s) = κ β_{1} (s) \dots β_{p + 1} (s)$ for some $κ \in ℂ$ , hence
$η_{1} + \dots + η_{q + 1} = n - \sum_{i = 1}^{p + 1} \deg (β_{i}) = n - \sum_{i = 1}^{p} \deg (α_{i}) - \deg (γ^{L}) = μ_{1} + \dots + μ_{p} + c_{1} + 1 - \deg (γ^{L}) .$

Next lemma provides a characterization of the invariant factors of matrices of the form (21).

Lemma 4.Let $σ_{1} (s) | \dots | σ_{p} (s)$ be monic polynomials and $M (s) = [\begin{matrix} γ (s) \\ M_{2} (s) \end{matrix}] = [\begin{array}{cccc} γ_{0} (s) & γ_{1} (s) & \dots & γ_{p} (s) \\ 0 & σ_{1} (s) & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & σ_{p} (s) \end{array}] \in ℂ {[s]}^{(1 + p) \times (1 + p)}$ a polynomial matrix with invariant factors $τ_{1} (s) | \dots | τ_{p + 1} (s)$ . Then, $γ_{0} (s) σ_{1} (s) \dots σ_{p} (s) = κ τ_{1} (s) \dots τ_{p + 1} (s)$ for some $κ \in ℂ$ , and for $1 \leq k \leq p$ , $τ_{1} (s) \dots τ_{k} (s)$ is the monic greatest common divisor of the polynomials in the following list.

1.
$σ_{1} (s) \dots σ_{k} (s)$ ,
2.
$γ_{0} (s) σ_{1} (s) \dots σ_{k - 1} (s)$ ,
3.
$γ_{i} (s) σ_{1} (s) \dots σ_{i - 1} (s) σ_{i + 1} (s) \dots σ_{k} (s), 1 \leq i \leq k$ .
4.
$γ_{i} (s) σ_{1} (s) \dots σ_{k - 1} (s), k + 1 \leq i \leq p$ ,

where we take

σ_{a} (s) \dots σ_{b} (s) = 1

for

a > b

Proof.The proof can be obtained taking into account that $\det (M (s)) = κ τ_{1} (s) \dots τ_{p + 1} (s)$ for some $κ \in ℂ$ , and that, for $1 \leq k \leq p$ , the minors of order k of $M (s)$ are multiples of one polynomial of the list (see the proof of Reference 16 [theorem 3.5]).

Lemma 5.Given a pencil $a (s) = [\begin{matrix} a_{1} (s) & \dots & a_{g} (s) \end{matrix}] \in 𝒫_{1 \times g} (ℂ)$ , let $γ (s) = \sum_{j = 1}^{g} s^{j - 1} a_{j} (s)$ and let $γ^{'} (s) \in ℂ [s]$ be a polynomial such that $\deg (γ^{'}) \leq g$ . Then, there exists a pencil $a^{'} (s) = [\begin{matrix} a_{1}^{'} (s) & \dots & a_{g}^{'} (s) \end{matrix}] \in 𝒫_{1 \times g} (ℂ)$ such that $γ^{'} (s) = \sum_{j = 1}^{g} s^{j - 1} a_{j}^{'} (s)$ and $‖ a^{'} (s) - a (s) ‖ = ‖ γ^{'} (s) - γ (s) ‖ .$

Proof.Let $a_{j} (s) = s a_{j, 1} + a_{j, 0}$ , $1 \leq j \leq g$ , $γ (s) = \sum_{j = 0}^{g} s^{j} γ_{j}$ , $γ^{'} (s) = \sum_{j = 0}^{g} s^{j} γ_{j}^{'}$ and $e_{j} = γ_{j}^{'} - γ_{j}$ , $0 \leq j \leq g$ . Then $γ_{0} = a_{1, 0}$ , $γ_{g} = a_{g, 1}$ , $γ_{j} = a_{j, 1} + a_{j + 1, 0}$ , $1 \leq j \leq g - 1$ , and $‖ γ^{'} (s) - γ (s) ‖ = \sum_{j = 0}^{g} | e_{j} | .$

Define $a_{j}^{'} (s) = s a_{j, 1}^{'} + a_{j, 0}^{'}$ , $1 \leq j \leq g$ , with $a_{1, 0}^{'} = a_{1, 0} + e_{0}$ , $a_{g, 1}^{'} = a_{g, 1} + e_{g}$ , and, for $1 \leq j \leq g - 1$ , $a_{j, 1}^{'} = a_{j, 1} + e_{j}$ , $a_{j + 1, 0}^{'} = a_{j + 1, 0}$ (or $a_{j, 1}^{'} = a_{j, 1}$ , $a_{j + 1, 0}^{'} = a_{j + 1, 0} + e_{j}$ ). Then $a^{'} (s) = [\begin{matrix} a_{1}^{'} (s) & \dots & a_{g}^{'} (s) \end{matrix}]$ satisfies the desired conditions.

Lemma 6.Let $\overline{M} (t) = [\begin{matrix} ā (t) & \overline{b} (t) & \overline{c} (t) \\ O & {\overline{M}}_{1} (t) & 0 \\ O & O & {\overline{M}}_{2} (t) \end{matrix}] \in ℂ {[t]}^{(1 + p + q) \times (1 + p + q)}$ and $\overline{N} (t) = [\begin{matrix} ā (t) & \overline{c} (t) \\ O & {\overline{M}}_{2} (t) \end{matrix}] \in ℂ {[t]}^{(1 + q) \times (1 + q)}$ be regular polynomial matrices such that ${\overline{M}}_{1} (0) = I_{p}$ . Then

w (0, \overline{M} (t)) = w (0, \overline{N} (t)) .

Proof.By³²^{(theorem 2)},

\sum_{i = 1}^{k} w_{i} (0, \overline{M} (t)) = (1 + p + q) k - rank ℛ_{k}, \sum_{i = 1}^{k} w_{i} (0, \overline{N} (t)) = (1 + q) k - rank 𝒮_{k}, k \geq 1,

with

\begin{align} ℛ_{k} = & [\begin{matrix} \overline{M} (0) & 0 & \dots & 0 & 0 \\ \frac{1}{1!} {\overline{M}}^{(1)} (0) & \overline{M} (0) & \dots & 0 & 0 \\ \frac{1}{2!} {\overline{M}}^{(2)} (0) & \frac{1}{1!} {\overline{M}}^{(1)} (0) & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ \frac{1}{(k - 1)!} {\overline{M}}^{(k - 1)} (0) & \frac{1}{(k - 2)!} {\overline{M}}^{(k - 2)} (0) & \dots & \frac{1}{1!} {\overline{M}}^{(1)} (0) & \overline{M} (0) \end{matrix}], 𝒮_{k} \\ = & [\begin{matrix} \overline{N} (0) & 0 & \dots & 0 & 0 \\ \frac{1}{1!} {\overline{N}}^{(1)} (0) & \overline{N} (0) & \dots & 0 & 0 \\ \frac{1}{2!} {\overline{N}}^{(2)} (0) & \frac{1}{1!} {\overline{N}}^{(1)} (0) & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ \frac{1}{(k - 1)!} {\overline{N}}^{(k - 1)} (0) & \frac{1}{(k - 2)!} {\overline{N}}^{(k - 2)} (0) & \dots & \frac{1}{1!} {\overline{N}}^{(1)} (0) & \overline{N} (0) \end{matrix}], \end{align}

where the superscript

^{(j)}

indicates the jth-derivative of the matrix with respect to t. Taking into account that

\overline{M} (0) = [\begin{matrix} ā (0) & \overline{b} (0) & \overline{c} (0) \\ O & I_{p} & 0 \\ O & O & {\overline{M}}_{2} (0) \end{matrix}]

, performing elementary operations it is easy to see that

rank ℛ_{k} = p k + rank 𝒮_{k}

, from where

\sum_{i = 1}^{k} w_{i} (0, \overline{M} (t)) = \sum_{i = 1}^{k} w_{i} (0, \overline{N} (t))

, for

k \geq 1

, and the lemma follows.

Lemma 7.Let $σ_{1} (s) | \dots | σ_{p} (s)$ , $τ_{1} (s) | \dots | τ_{p + 1} (s)$ , $τ_{1}^{'} (s) | \dots | τ_{p + 1}^{'} (s)$ be monic polynomials such that

\begin{align} τ_{i}^{'} (s) | σ_{i} (s) | τ_{i + 1}^{'} (s), 1 \leq i \leq p, \end{align}

()

and

\begin{align} (τ_{1}^{'} (s), \dots, τ_{p + 1}^{'} (s)) ≺ (τ_{1} (s), \dots, τ_{p + 1} (s)) . \end{align}

()

Let

γ_{0} (s), γ_{1} (s), \dots, γ_{p} (s) \in ℂ [s]

be polynomials such that

\deg (γ_{i}) \leq \deg (σ_{i})

1 \leq i \leq p

, and let

M (s) = [\begin{array}{cccc} γ_{0} (s) & γ_{1} (s) & \dots & γ_{p} (s) \\ 0 & σ_{1} (s) & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & σ_{p} (s) \end{array}] \in ℂ {[s]}^{(1 + p) \times (1 + p)}

be a matrix with invariant factors

τ_{1} (s) | \dots | τ_{p + 1} (s)

. Then, for every

ϵ > 0

there exists

M^{'} (s) = [\begin{array}{cccc} γ_{0} (s) & γ_{1}^{'} (s) & \dots & γ_{p}^{'} (s) \\ 0 & σ_{1} (s) & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & σ_{p} (s) \end{array}] \in ℂ {[s]}^{(1 + p) \times (1 + p)}

with invariant factors

τ_{1}^{'} (s) | \dots | τ_{p + 1}^{'} (s)

, such that

\deg (γ_{i}^{'}) \leq \deg (σ_{i})

1 \leq i \leq p

, and

‖ M^{'} (s) - M (s) ‖ < ϵ

Proof.The invariant factors of the polynomial matrix $[\begin{array}{cccc} 0 & σ_{1} (s) & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & σ_{p} (s) \end{array}] \in ℂ {[s]}^{p \times (1 + p)}$ are $σ_{1} (s) | \dots | σ_{p} (s)$ . By Theorem 4,

τ_{i} (s) | σ_{i} (s) | τ_{i + 1} (s), 1 \leq i \leq p .

Now the proof follows the steps of that of Theorem 3.5 in Reference 16.

In the next theorem, we solve the announced problem of prescription of the homogeneous invariant factors.

Theorem 8.Let $ψ_{1}^{'} (s, t) | \dots | ψ_{n}^{'} (s, t)$ be homogeneous polynomials, monic with respect to s. For every $ϵ > 0$ , there exists a pencil $A^{'} (s) = [\begin{matrix} a^{'} (s) \\ A_{2} (s) \end{matrix}] \in 𝒫_{(1 + (n - 1)) \times n} (ℂ)$ with $ψ_{1}^{'} (s, t) | \dots | ψ_{n}^{'} (s, t)$ as homogeneous invariant factors such that $‖ A^{'} (s) - A (s) ‖ < ϵ$ if and only if (13) and (18) hold.

Proof.The necessity of the conditions follows from Theorem 7. Let us prove the sufficiency.

Since $A_{2} (s)$ has p nontrivial invariant factors $α_{1} (s) | \dots | α_{p}$ and q nontrivial infinite elementary divisors $t^{μ_{1}} | \dots | t^{μ_{q}}$ , by Theorem 5, the pencil $A (s)$ has at most $p + 1$ nontrivial invariant factors $β_{1} (s) | \dots | β_{p + 1} (s)$ and $q + 1$ nontrivial infinite elementary divisors $s^{η_{1}} | \dots | s^{η_{q + 1}}$ satisfying

β_{i} (s) | α_{i} (s) | β_{i + 1} (s), 1 \leq i \leq p, and η_{i} \leq μ_{i} \leq η_{i + 1}, 1 \leq i \leq q .

Denoting

{\hat{β}}_{i}^{'} (s) = ψ_{i}^{'} (s, 1) and ψ_{i}^{'} (s, t) = t^{{\hat{η}}_{i}^{'}} t^{\deg ({\hat{β}}_{i}^{'})} {\hat{β}}_{i}^{'} (\frac{s}{t}), 1 \leq i \leq n,

from (18)

{\hat{β}}_{i}^{'} (s) = 1, 1 \leq i \leq n - p - 1, and {\hat{η}}_{i}^{'} = 0, 1 \leq i \leq n - q - 1,

and, taking

β_{i}^{'} (s) = {\hat{β}}_{n - p - 1 + i}^{'} (s), 1 \leq i \leq p + 1

η_{i}^{'} = {\hat{η}}_{n - q - 1 + i}^{'}, 1 \leq i \leq q + 1

, from (18) and (13) we obtain

\begin{align} β_{i}^{'} (s) | α_{i} (s) | β_{i + 1}^{'} (s), 1 \leq i \leq p, and (β_{1}^{'} (s), \dots, β_{p + 1}^{'} (s)) ≺ (β_{1} (s), \dots, β_{p + 1} (s)), \end{align}

()

\begin{align} η_{i}^{'} \leq μ_{i} \leq η_{i + 1}^{'}, 1 \leq i \leq q, and (t^{η_{i}^{'}}, \dots, t^{η_{q + 1}^{'}}) ≺ (t^{η_{1}}, \dots, t^{η_{q + 1}}) . \end{align}

()

We partition $a (s)$ as in Lemma 3. According to this lemma, $A (s)$ and ${rev}_{1} (A) (t)$ are equivalent to $diag (I_{n - 1 - p}, M (s))$ and $diag (I_{n - 1 - p - q}, \overline{M} (t))$ , respectively, where $M (s)$ and $\overline{M} (t)$ are defined in (21) and (22).

Let $ϵ > 0$ . By Lemma 7, from (27) there exists $M^{'} (s) = [\begin{array}{cccc} γ^{L} (s) & {\overline{γ}}_{1}^{C} (s) & \dots & {\overline{γ}}_{p}^{C} (s) \\ 0 & α_{1} (s) & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & α_{p} (s) \end{array}] \in ℂ {[s]}^{(p + 1) \times (p + 1)}$ with invariant factors $β_{1}^{'} (s) | \dots β_{p + 1}^{'} (s)$ , such that $\deg ({\overline{γ}}_{i}^{C}) \leq g_{i}$ , $1 \leq i \leq p$ , and $‖ M^{'} (s) - M (s) ‖ < \frac{ϵ}{2}$ . Moreover, by Lemma 5, there exists a pencil

\begin{align} ā^{C} (s) = [\begin{array}{ccc} ā_{1, 1}^{C} (s) . . . ā_{1, g_{1}}^{C} (s) | \dots | ā_{p, 1}^{C} (s) . . . ā_{p, g_{p}}^{C} (s) \end{array}] \in 𝒫_{1 \times \sum_{i = 1}^{p} g_{i}} (ℂ), \end{align}

such that

‖ ā^{C} (s) - a^{C} (s) ‖ < \frac{ϵ}{2}

and

\begin{align} {\overline{γ}}_{i}^{C} (s) = \sum_{j = 1}^{g_{i}} s^{j - 1} {ā_{i, j}}^{C} (s), 1 \leq i \leq p . \end{align}

()

Let $\overline{N} (t) = [\begin{array}{cccc} {rev}_{c_{1} + 1} (γ^{L}) (t) & {rev}_{μ_{1}} (γ_{1}^{N}) (t) & \dots & {rev}_{μ_{q}} (γ_{q}^{N}) (t) \\ 0 & t^{μ_{1}} & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & t^{μ_{q}} \end{array}],$ and let $δ_{1} (t) | \dots | δ_{q + 1} (t)$ be its invariant factors. Let $g_{0} = \deg (γ^{L})$ . By Remark 4 item 3, $κ δ_{1} (t) \dots δ_{q + 1} (t) = t^{μ_{1} + \dots + μ_{q} + c_{1} + 1 - g_{0}} {\tilde{γ}}^{L} (t) = t^{η_{1} + \dots + η_{q + 1}} {\tilde{γ}}^{L} (t)$ for some $κ \in ℂ$ . By Lemma 6, $w (0, \overline{N} (t)) = w (0, \overline{M} (t))$ . As $diag (I_{n - 1 - p - q}, \overline{M} (t))$ and ${rev}_{1} (A) (t)$ are equivalent, we have $w (0, \overline{M} (t)) = w (0, {rev}_{1} (A) (t)) = w (\infty, A (s)) = \overline{(η_{1}, \dots, η_{q + 1})}$ , and by Theorem 4, $δ_{i} (t) | t^{μ_{i}}$ , $1 \leq i \leq q$ . Therefore, $δ_{i} (t) = t^{η_{i}}$ , $1 \leq i \leq q$ and $κ δ_{q + 1} (t) = t^{η_{q + 1}} {\tilde{γ}}^{L} (t)$ .

We define $δ'_{i} (t) = t^{η_{i}^{'}}$ , $1 \leq i \leq q$ and $δ'_{q + 1} (t) = \frac{1}{κ} t^{η_{q + 1}^{'}} {\tilde{γ}}^{L} (t)$ . Then, from (28),

\begin{align} δ_{i}^{'} (t) | t^{μ_{i}} | δ_{i + 1}^{'} (t), 1 \leq i \leq q, (δ_{1}^{'} (t), \dots, δ_{q + 1}^{'} (t)) ≺ (δ_{1} (t), \dots, δ_{q + 1} (t)) . \end{align}

()

By Lemma 7, from (30), there exists

{\overline{N}}^{'} (t) = [\begin{array}{cccc} rev_{c_{1} + 1} (γ^{L}) (t) & γ_{1}^{'} (t) & \dots & γ_{q}^{'} (t) \\ 0 & t^{μ_{1}} & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & t^{μ_{q}} \end{array}] \in ℂ {[t]}^{(q + 1) \times (q + 1)},

with invariant factors

δ_{1}^{'} (t) | \dots | δ_{q + 1}^{'} (s)

, such that

\deg ({γ^{'}}_{i}) \leq μ_{i}

1 \leq i \leq q

, and

‖ \overline{N} (t) - {\overline{N}}^{'} (t) ‖ < \frac{ϵ}{2}

. Observe that

w (0, {\overline{N}}^{'} (t)) = \overline{(η_{1}^{'}, \dots, η_{q + 1}^{'})}

Let ${\overline{γ}}_{1}^{N} (s), \dots, {\overline{γ}}_{q}^{N} (s) \in ℂ [s]$ be polynomials such that ${rev}_{μ_{i}} ({\overline{γ}}_{i}^{N}) (t) = γ_{i}^{'} (t)$ . Recall that $‖ {\overline{γ}}_{i}^{N} (s) - γ_{i}^{N} (s) ‖ = ‖ {rev}_{μ_{i}} ({\overline{γ}}_{i}^{N}) (t) - {rev}_{μ_{i}} (γ_{i}^{N}) (t) ‖ .$ By Lemma 5, there exists a pencil

ā^{N} (s) = [\begin{array}{ccc} ā_{1, 1}^{N} (s) . . . ā_{1, μ_{1}}^{N} (s) | \dots | ā_{q, 1}^{N} (s) . . . ā_{q, μ_{q}}^{N} (s) \end{array}] \in 𝒫_{1 \times \sum_{i = 1}^{q} μ_{i}} (ℂ),

such that

‖ ā^{N} (s) - a^{N} (s) ‖ < \frac{ϵ}{2}

and

{\overline{γ}}_{i}^{N} (s) = \sum_{j = 1}^{μ_{i}} s^{μ_{i} - j} ā_{i, j}^{N} (s), 1 \leq i \leq q .

()

Let

{\overline{M}}^{'} (t) = [\begin{array}{ccccccc} rev_{c_{1} + 1} ({\overline{γ}}^{L}) (t) & rev_{g_{1}} ({\overline{γ}}_{1}^{C}) (t) & \dots & rev_{g_{p}} ({\overline{γ}}_{p}^{C}) (t) & rev_{μ_{1}} ({\overline{γ}}_{1}^{N}) (t) & \dots & rev_{μ_{q}} ({\overline{γ}}_{q}^{N}) (t) \\ 0 & {\tilde{α}}_{1} (t) & \dots & 0 & 0 & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & {\tilde{α}}_{p} (t) & 0 & \dots & 0 \\ 0 & 0 & \dots & 0 & t^{μ_{1}} & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & 0 & 0 & \dots & t^{μ_{q}} \end{array}] .

By Lemma 6,

w (0, {\overline{M}}^{'} (t)) = w (0, {\overline{N}}^{'} (t)) = \overline{(η_{1}^{'}, \dots, η_{q + 1}^{'})}

Let $a^{'} (s) = [\begin{matrix} a^{L} (s) & ā^{C} (s) & ā^{N} (s) \end{matrix}]$ and $A^{'} (s) = [\begin{matrix} a^{'} (s) \\ A_{2} (s) \end{matrix}]$ . Then $‖ A^{'} (s) - A (s) ‖ < ϵ$ . By Lemma 3, $A^{'} (s)$ and ${rev}_{1} (A^{'}) (t)$ are equivalent to $diag (I_{n - 1 - p}, M^{'} (s))$ and $diag (I_{n - 1 - p - q}, {\overline{M}}^{'} (t))$ , respectively. Therefore, the invariant factors of $A^{'} (s)$ are

{\hat{β}}_{i}^{'} (s) = 1, 1 \leq i \leq n - p - 1, {\hat{β}}_{n - p - 1 + i}^{'} (s) = β_{i}^{'} (s), 1 \leq i \leq p + 1 .

Moreover,

w (\infty, A^{'} (s)) = w (0, {\overline{M}}^{'} (t)) = \overline{(η_{1}^{'}, \dots, η_{q + 1}^{'})},

therefore, the homogeneous invariant factors of

A^{'} (s)

are

ψ_{1}^{'} (s, t) | \dots | ψ_{n}^{'} (s, t)

In order to prescribe the Weyr characteristic of $A^{'} (s)$ we will use some auxiliary lemmas. First of all, we state Lemma 4 in terms of the partial multiplicities of the eigenvalues of $M (s)$ .

Lemma 8.Let $M (s) = [\begin{matrix} γ (s) \\ M_{2} (s) \end{matrix}] \in ℂ {[s]}^{(1 + p) \times (1 + p)}$ be the matrix in Lemma 4, let $λ \in ℂ$ and write

γ_{i} (s) = {(s - λ)}^{x_{i}} {\hat{γ}}_{i} (s), {\hat{γ}}_{i} (λ) \neq 0, x_{i} \geq 0, 0 \leq i \leq p .

Then,

\sum_{i = 1}^{p + 1} n_{i} (λ, M (s)) = x_{0} + \sum_{i = 1}^{p} n_{i} (λ, M_{2} (s))

and for

2 \leq ℓ \leq p + 1

\sum_{i = ℓ}^{p + 1} n_{i} (λ, M (s))

is the minimum of the integers in the following list.

1.
$\sum_{i = ℓ - 1}^{p} n_{i} (λ, M_{2} (s))$ ,
2.
$x_{0} + \sum_{i = ℓ}^{p} n_{i} (λ, M_{2} (s))$ ,
3.
$x_{j} + \sum_{i = p - j + 2}^{p} n_{i} (λ, M_{2} (s)) + \sum_{i = ℓ - 1}^{p - j} n_{i} (λ, M_{2} (s)), 1 \leq j \leq p - ℓ + 2$ ,
4.
$x_{j} + \sum_{i = ℓ}^{p} n_{i} (λ, M_{2} (s)), p - ℓ + 3 \leq j \leq p$ ,

where we take $\sum_{i = a}^{b} n_{i} (λ, M_{2} (s)) = 0$ for $a > b$ .

Observe that Theorem 4 implies, for

λ \in ℂ

n_{i + 1} (λ, M (s)) \leq n_{i} (λ, M_{2} (s)) \leq n_{i} (λ, M (s)), 1 \leq i \leq p,

hence

x_{0} \geq n_{1} (λ, M (s)) - n_{1} (λ, M_{2} (s))

Lemma 9.Under the notation of Lemmas 4 and 8, let z be an integer, $0 \leq z \leq n_{1} (λ, M (s)) - n_{1} (λ, M_{2} (s))$ , $γ_{0}^{'} (s) = {(s - λ)}^{x_{0} - z} {\hat{γ}}_{0}^{'} (s),$ with ${\hat{γ}}_{0}^{'} (λ) \neq 0,$ and $M^{'} (s) = [\begin{matrix} γ^{'} (s) \\ M_{2} (s) \end{matrix}] = [\begin{array}{cccc} γ_{0}^{'} (s) & γ_{1} (s) & \dots & γ_{p} (s) \\ 0 & σ_{1} (s) & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & σ_{p} (s) \end{array}]$ . Then

\begin{align} n_{1} (λ, M^{'} (s)) = n_{1} (λ, M (s)) - z and n_{k} (λ, M^{'} (s)) = n_{k} (λ, M (s)), 2 \leq k \leq p + 1 . \end{align}

()

Proof.Notice that for $M^{'} (s)$ , the values of the expressions in items 1, 3, and 4 in Lemma 8 coincide with those of $M (s)$ , whereas the expression in item 2 turns into $x_{0} - z + \sum_{i = ℓ}^{p} n_{i} (λ, M_{2} (s))$ for $M^{'} (s)$ .

From Lemma 8, we obtain that $\sum_{i = 1}^{p + 1} n_{i} (λ, M^{'} (s)) = x_{0} - z + \sum_{i = 1}^{p} n_{i} (λ, M_{2} (s)) = \sum_{i = 1}^{p + 1} n_{i} (λ, M (s)) - z,$ and $\sum_{i = k}^{p + 1} n_{i} (λ, M^{'} (s)) \leq \sum_{i = k}^{p + 1} n_{i} (λ, M (s))$ , $2 \leq k \leq p + 1$ .

Assume that for some $k \in {2, \dots, p + 1}$ , $\sum_{i = k}^{p + 1} n_{i} (λ, M^{'} (s)) < \sum_{i = k}^{p + 1} n_{i} (λ, M (s))$ . Then

\begin{align} \sum_{i = k}^{p + 1} n_{i} (λ, M^{'} (s)) & = x_{0} - z + \sum_{i = k}^{p} n_{i} (λ, M_{2} (s)) = \sum_{i = 1}^{p + 1} n_{i} (λ, M (s)) - \sum_{i = 1}^{p} n_{i} (λ, M_{2} (s)) - z + \sum_{i = k}^{p} n_{i} (λ, M_{2} (s)) \\ = \sum_{i = 1}^{k - 1} (n_{i} (λ, M (s)) - n_{i} (λ, M_{2} (s)) - z + \sum_{i = k}^{p + 1} n_{i} (λ, M (s)) . \end{align}

Bearing in mind that

n_{i} (λ, M (s)) \geq n_{i} (λ, M_{2} (s))

1 \leq i \leq p

and

n_{1} (λ, M (s)) - n_{1} (λ, M_{2} (s)) \geq z

, we obtain that

\sum_{i = k}^{p + 1} n_{i} (λ, M^{'} (s)) \geq \sum_{i = k}^{p + 1} n_{i} (λ, M (s))

, which is a contradiction. Therefore, for

2 \leq k \leq p + 1

\sum_{i = k}^{p + 1} n_{i} (λ, M^{'} (s)) = \sum_{i = k}^{p + 1} n_{i} (λ, M (s))

, hence (32) is satisfied.

The proof of the following lemma can be found in Reference 33 (lemma 3.2).

Lemma 10.Let $(a_{1}, \dots)$ and $(b_{1}, \dots)$ be partitions of nonnegative integers. Let $p = (p_{1}, \dots) = \overline{(a_{1}, \dots)}$ and $q = (q_{1}, \dots) = \overline{(b_{1}, \dots)}$ be the conjugate partitions. Let $k \geq 0$ be an integer. Then, $a_{i} \geq b_{i + k},$ $i \geq 1,$ if and only if $p_{i} \geq q_{i} - k,$ $i \geq 1$ .

In the next theorem, given a matrix pencil $A (s)$ , we provide conditions that some prescribed partitions must satisfy in order to be the Weyr characteristic of a pencil obtained from $A (s)$ by a small perturbation of one row. Recall that the pencil $A (s)$ is split as $A (s) = [\begin{matrix} a (s) \\ A_{2} (s) \end{matrix}] \in 𝒫_{(1 + (n - 1)) \times n} (ℂ)$ .

Theorem 9.Let $𝒱_{η} (A (s))$ be an $η$ -neighborhood of the spectrum of $A (s)$ . For each $λ \in Λ (A (s))$ let $t_{λ}$ be a given integer $t_{λ} \geq 1$ and let $w^{(λ, j)} = (w_{1}^{(λ, j)}, \dots)$ be given partitions, $1 \leq j \leq t_{λ}$ , such that $w^{(λ, j)} \neq (0)$ , $2 \leq j \leq t_{λ}$ .

For every $ϵ > 0$ , there exists a pencil $A^{'} (s) = [\begin{matrix} a^{'} (s) \\ A_{2} (s) \end{matrix}] \in 𝒫_{(1 + (n - 1)) \times n} (ℂ)$ such that $‖ A^{'} (s) - A (s) ‖ < ϵ$ , the spectrum of $A^{'} (s)$ satisfies condition (4), and

\begin{array}{r} w (λ, A^{'} (s)) = w^{(λ, 1)}, \\ A^{'} (s) has t_{λ} - 1 eigenvalues μ_{λ, 2}, \dots, μ_{λ, t_{λ}} in B (λ, η), different from λ, with w (μ_{λ, j}, A^{'} (s)) = w^{(λ, j)}, 2 \leq j \leq t_{λ}, \end{array}\}

()

if and only if condition (7) and

\begin{array}{r} 0 \leq w_{i}^{(λ, 1)} - w_{i} (λ, A_{2} (s)) \leq 1, i \geq 1, \\ 0 \leq w_{i}^{(λ, j)} \leq 1, i \geq 1, 2 \leq j \leq t_{λ}, \end{array}\}

()

are satisfied.

Proof.The proof is inspired by that of Reference 16 (theorem 3.8). Assume that for every $ϵ > 0$ , there exists a pencil $A^{'} (s) = [\begin{matrix} a^{'} (s) \\ A_{2} (s) \end{matrix}] \in 𝒫_{(1 + (n - 1)) \times n} (ℂ)$ satisfying (4) and (33) and such that $‖ A^{'} (s) - A (s) ‖ < ϵ$ . Then, from Theorem 2, condition (7) holds, and from Theorem 8 we obtain (18), which is equivalent to

\begin{align} n_{i + 1} (μ, A^{'} (s)) \leq n_{i} (μ, A_{2} (s)) \leq n_{i} (μ, A^{'} (s)), i \geq 1, for each μ \in \overline{ℂ} . \end{align}

()

For

λ \in Λ (A (s))

and

2 \leq j \leq t_{λ}

μ_{λ, j} \notin Λ (A (s))

, hence

n (μ_{λ, j}, A (s)) = n (μ_{λ, j}, A_{2} (s)) = (0)

. Then, by Lemma 10, (35) implies (34).

Conversely, assume that (7) and (34) hold, and let $ϵ > 0$ . Recall that $A_{2} (s)$ is in the Kronecker canonical form given in (19). For each $λ \in Λ (A (s))$ and $1 \leq j \leq t_{λ}$ , let $n^{(λ, j)} = \overline{w^{(λ, j)}} = (n_{1}^{(λ, j)}, \dots)$ . Then, from (34) and Lemma 10 we obtain

\begin{align} n_{i + 1}^{(λ, 1)} \leq n_{i} (λ, A_{2} (s)) \leq n_{i}^{(λ, 1)}, i \geq 1, \end{align}

()

and

\begin{align} n_{2}^{(λ, j)} = 0, 2 \leq j \leq t_{λ} . \end{align}

()

For

λ \in Λ (A (s))

we denote

p^{λ} = \sum_{j = 1}^{t_{λ}} n^{(λ, j)} = (p_{1}^{λ}, \dots)

and define

\begin{align} ξ_{n - i + 1} (s, t) = t^{p_{i}^{\infty}} \prod_{λ \in Λ (A (s) ∖ {\infty}} {(s - λ t)}^{p_{i}^{λ}}, 1 \leq i \leq n, \end{align}

()

where, if

\infty \notin Λ (A (s)

, we take

p^{\infty} = (0)

. Then, (7) and (36) are equivalent to

\begin{align} (ξ_{1} (s, t), \dots, ξ_{n} (s, t)) ≺ (ψ_{1} (s, t), \dots, ψ_{n} (s, t)) and ξ_{i} (s, t) | ϕ_{i} (s, t) | ξ_{i + 1} (s, t), 1 \leq i \leq n - 1, \end{align}

respectively. By Theorem 8, there exists a pencil

B (s) = [\begin{matrix} b (s) \\ A_{2} (s) \end{matrix}] \in 𝒫_{(1 + (n - 1)) \times n} (ℂ)

with

ξ_{1} (s, t) | \dots | ξ_{n} (s, t)

as homogeneous invariant factors and such that

‖ B (s) - A (s) ‖ < \frac{ϵ}{2}

. Observe that

n (λ, B (s)) = p^{λ}

for each

λ \in Λ (A (s))

We partition $b (s)$ according to the blocks of $A_{2} (s)$ ,

\begin{align} b (s) & = [\begin{matrix} b^{L} (s) & b^{C} (s) & b^{N} (s) \end{matrix}] \\ = [\begin{array}{c} \begin{array}{c} b_{1}^{L} (s) . . . b_{c_{1} + 1}^{L} (s) ||b_{1, 1}^{C} (s . . . b_{1, g_{1}}^{C} (s) | . . . | b_{p, 1}^{C} (s) . . . b_{p, g_{p}}^{C} (s)|| b_{1, 1}^{N} (s) . . . b_{1, μ_{1}}^{N} (s) | . . . | \end{array} b_{q, 1}^{N} (s) . . . b_{q, μ_{q}}^{N} (s) \end{array}] . \end{align}

By Lemma 3,

B (s)

and

{rev}_{1} (B) (t)

are equivalent to

diag (I_{n - 1 - p}, N (s))

and

diag (I_{n - 1 - p - q}, \overline{N} (t))

, respectively, where

\begin{align} N (s) & = [\begin{matrix} θ (s) \\ M_{2} (s) \end{matrix}] = [\begin{array}{cccc} θ^{L} (s) & θ_{1}^{C} (s) & \dots & θ_{p}^{C} (s) \\ 0 & α_{1} (s) & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & α_{p} (s) \end{array}], \\ \overline{N} (t) & = [\begin{array}{ccccccc} rev_{c_{1} + 1} (θ^{L}) (t) & rev_{g_{1}} (θ_{1}^{C}) (t) & \dots & rev_{g_{p}} (θ_{p}^{C}) (t) & rev_{μ_{1}} (θ_{1}^{N}) (t) & \dots & rev_{μ_{q}} (θ_{q}^{N}) (t) \\ 0 & {\tilde{α}}_{1} (t) & \dots & 0 & 0 & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & {\tilde{α}}_{p} (t) & 0 & \dots & 0 \\ 0 & 0 & \dots & 0 & t^{μ_{1}} & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & 0 & 0 & \dots & t^{μ_{q}} \end{array}], \end{align}

with

\begin{align} θ^{L} (s) = \sum_{j = 1}^{c_{1} + 1} s^{j - 1} b_{j}^{L} (s), θ_{i}^{C} (s) = \sum_{j = 1}^{g_{i}} s^{j - 1} b_{i, j}^{C} (s), 1 \leq i \leq p, θ_{i}^{N} (s) = \sum_{j = 1}^{μ_{i}} s^{μ_{i} - j} b_{i, j}^{N} (s), 1 \leq i \leq q . \end{align}

Then,

N (s)

has the same nontrivial invariant factors as

B (s)

, hence

α_{1} (s) \dots α_{p} (s) θ^{L} (s) = κ ψ_{1} (s, 1) \dots ψ_{n} (s, 1) = κ ξ_{1} (s, 1) \dots ξ_{n} (s, 1)

for some

κ \in ℂ

. Therefore

θ^{L} (s) = κ \prod_{λ \in Λ (A (s) ∖ {\infty}} {(s - λ)}^{x_{0, λ}},

where, for

λ \in Λ (A (s)

x_{0, λ} = \sum_{i = 1}^{n} n_{i} (λ, B (s)) - \sum_{i = 1}^{n - 1} n_{i} (λ, A_{2} (s)) = \sum_{i = 1}^{n} n_{i} (λ, A (s)) - \sum_{i = 1}^{n - 1} n_{i} (λ, A_{2} (s))

. Observe that

\deg (θ^{L}) = \deg (ψ_{1} (s, 1) \dots ψ_{n} (s, 1)) - \deg (ϕ_{1} (s, 1) \dots ϕ_{n - 1} (s, 1)) = c_{1} + 1 + \sum_{i = 1}^{n - 1} n_{i} (\infty, A_{2} (s)) - \sum_{i = 1}^{n} n_{i} (\infty, A (s)) = c_{1} + 1 - x_{0, \infty}

, thus

{rev}_{c_{1} + 1} (θ^{L}) (t) = t^{c_{1} + 1 - \deg (θ^{L})} {\tilde{θ}}^{L} (t) = t^{x_{0, \infty}} {\tilde{θ}}^{L} (t) .

For each $λ \in Λ (A (s))$ and $2 \leq j \leq t_{λ}$ , let $μ_{λ, j} \in ℂ$ with $μ_{λ, j} \neq λ$ , and $z_{λ} = \sum_{j = 2}^{t_{λ}} n_{1}^{(λ, j)}$ . We have

n_{1} (λ, N (s)) - n_{1} (λ, M_{2} (s)) = n_{1} (λ, B (s)) - n_{1} (λ, A_{2} (s)) = p_{1}^{λ} - n_{1} (λ, A_{2} (s)) = n_{1}^{(λ, 1)} - n_{1} (λ, A_{2} (s)) + z_{λ} .

By (36), we obtain

z_{λ} \leq n_{1} (λ, N (s)) - n_{1} (λ, M_{2} (s)) \leq x_{0, λ}

Let

\begin{align} {\overline{θ}}_{1}^{L} (s) = \prod_{λ \in Λ (A (s) ∖ {\infty}} ({(s - λ)}^{x_{0, λ} - z_{λ}} \prod_{j = 2}^{t_{λ}} {(s - μ_{λ, j})}^{n_{1}^{(λ, j)}}), {\overline{θ}}^{L} (s) = \prod_{j = 2}^{t_{\infty}} {(1 - (\frac{1}{μ_{\infty, j}}) s)}^{n_{1}^{(\infty, j)}} {\overline{θ}}_{1}^{L} (s), \end{align}

()

and

\begin{align} N^{'} (s) & = [\begin{matrix} θ^{'} (s) \\ M_{2} (s) \end{matrix}] = [\begin{array}{cccc} {\overline{θ}}^{L} (s) & θ_{1}^{C} (s) & \dots & θ_{p}^{C} (s) \\ 0 & α_{1} (s) & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & α_{p} (s) \end{array}], \\ {\overline{N}}^{'} (t) & = [\begin{array}{ccccccc} rev_{c_{1} + 1} ({\overline{θ}}^{L}) (t) & rev_{g_{1}} (θ_{1}^{C}) (t) & \dots & rev_{g_{p}} (θ_{p}^{C}) (t) & rev_{μ_{1}} (θ_{1}^{N}) (t) & \dots & rev_{μ_{q}} ({\overline{θ}}_{q}^{N}) (t) \\ 0 & {\tilde{α}}_{1} (t) & \dots & 0 & 0 & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & {\tilde{α}}_{p} (t) & 0 & \dots & 0 \\ 0 & 0 & \dots & 0 & t^{μ_{1}} & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & 0 & 0 & \dots & t^{μ_{q}} \end{array}] . \end{align}

Observe that for each

λ \in Λ (A (s))

and

2 \leq j \leq t_{λ}

n_{1} (μ_{λ, j}, N^{'} (s)) = n_{1}^{(λ, j)}, n_{k} (μ_{λ, j}, N^{'} (s)) = 0 = n_{k}^{(λ, j)}, 2 \leq k \leq p + 1,

that is,

w (μ_{λ, j}, N^{'} (s)) = w^{(λ, j)}

2 \leq j \leq t_{λ}

. Moreover,

\deg ({\overline{θ}}^{L}) = \deg (θ^{L}) + z_{\infty}

, hence

\begin{align} {rev}_{c_{1} + 1} ({\overline{θ}}^{L}) (t) = t^{c_{1} + 1 - \deg ({\overline{θ}}^{L})} {\tilde{\overline{θ}}}^{L} (t) = t^{x_{0, \infty} - z_{\infty}} {\tilde{\overline{θ}}}^{L} (t) . \end{align}

()

\deg ({\overline{θ}}^{L}) = c_{1} + 1 - (x_{0, \infty} - z_{\infty}) \leq c_{1} + 1

, by Lemma 9, from (39) and (40), for

λ \in Λ (A (s))

n_{1} (λ, N^{'} (s)) = n_{1} (λ, N (s)) - z_{λ} = n_{1}^{(λ, 1)}, and n_{k} (λ, N^{'} (s)) = n_{k} (λ, N (s)) = n_{k}^{(λ, 1)}, 2 \leq k \leq p + 1,

that is,

w (λ, N^{'} (s)) = w^{(λ, 1)}

. By Lemma 5, there exists a pencil

{\overline{b}}^{L} (s) \in 𝒫_{1 \times (c_{1} + 1)}

such that

‖ {\overline{b}}^{L} (s) - b^{L} (s) ‖ = ‖ {\overline{θ}}^{L} (s) - θ^{L} (s) ‖

and if

B^{'} (s) = [\begin{matrix} b^{'} (s) \\ A_{2} (s) \end{matrix}] \in 𝒫_{(1 + (n - 1)) \times n} (ℂ)

with

b^{'} (s) = [\begin{matrix} {\overline{b}}^{L} (s) & b^{C} (s) & b^{N} (s) \end{matrix}]

, then

B^{'} (s)

and

{rev}_{1} (B^{'}) (t)

are equivalent to

diag (I_{n - 1 - p}, N^{'} (s))

and

diag (I_{n - 1 - p - q}, {\overline{N}}^{'} (t))

, respectively. Hence, for each

λ \in Λ (A (s))

w (λ, B^{'} (s)) = w (λ, N^{'} (s)) = w^{(λ, 1)}

and

w (μ_{λ, j}, B^{'} (s)) = w (μ_{λ, j}, N^{'} (s)) = w^{(λ, j)}

2 \leq j \leq t_{λ}

By Lemma 2, for $λ \in Λ (A (s) ∖ {\infty}$ , the values of $μ_{λ, j}$ can be chosen in $B (λ, η)$ , $2 \leq j \leq t_{λ}$ , in such a way that $‖ {\overline{θ}}_{1}^{L} (s) - θ^{L} (s) ‖ < \frac{ϵ}{4}$ , and it is easy to see that we can choose $μ_{\infty, j} \in B (\infty, η)$ such that $‖ {\overline{θ}}^{L} (s) - {\overline{θ}}_{1}^{L} (s) ‖ < \frac{ϵ}{4}$ , $2 \leq j \leq t_{\infty}$ . Therefore, $‖ B^{'} (s) - B (s) ‖ = ‖ {\overline{b}}^{L} (s) - b^{L} (s) ‖ = ‖ {\overline{θ}}^{L} (s) - θ^{L} (s) ‖ < \frac{ϵ}{2}$ , and as a consequence we obtain $‖ B^{'} (s) - A (s) ‖ < ϵ .$

5 CONCLUSIONS AND FUTURE RESEARCH

The effect of small perturbations of a regular pencil when some of its rows remain unchanged is investigated. It is a twofold problem. On one hand, it involves characteristics of general small perturbation problems. On the other hand, it is closely related to matrix pencil completion problems. We have obtained necessary conditions to be satisfied by the Weierstrass invariants of a pencil which is a one-row small perturbation of another regular pencil (see Theorems 6 and 7).

Moreover, when perturbing a single row, we also prove the sufficiency of the necessary conditions obtained (see Theorems 8 and 9). Our results generalize to pencils previous results on the problem obtained for square matrices. To achieve them, we had to tackle the difficulties appearing due to the presence of infinite elementary divisors in the pencils.

Our next step is to extend the sufficiency part of this work to regular pencils when more than one row is perturbed. The research can also be extended to singular pencils.

ACKNOWLEDGMENT

This work has been partially supported by “Ministerio de Economía, Industria y Competitividad (MINECO)” of Spain through grant MTM2017-83624-P and “Fondo Europeo de Desarrollo Regional (FEDER)” of EU through grant MTM2017-83624-P.

CONFLICT OF INTEREST

The authors declare no potential conflict of interests.

Biographies

Itziar Baragaña Ph.D. in Mathematical Sciences by the University of the Basque Country (UPV/EHU). She is Associate Professor in the Department of Computer Science and Artificial Intelligence Department, UPV/EHU. She is also a member of the Research group GAMA of the University of the Basque Country (UPV/EHU). Her research interest includes linear algebra, matrix theory, and linear systems and control theory.
Alicia Roca Ph.D. in Mathematical Sciences by the Universitat Politècnica de València (UPV), Valencia, Spain. She is Associate Professor in the Department of Matemática Aplicada, at the E.T.S. Telecommunications Engineering, UPV, and a member of the IMM Institute (UPV). She is also a member of the Research group GAMA of the University of the Basque Country (UPV/EHU). Her research interests are linear algebra, matrix theory and applications, and cryptography.

REFERENCES

1Mehl C, Mehrmann V, Ran A, Rodman L. Eigenvalue perturbation theory of classes of structured matrices under generic structured rank one perturbations. Linear Algebra Appl. 2011; 435: 687-716.
10.1016/j.laa.2010.07.025
Web of Science® Google Scholar
2Silva F. The rank of the difference of matrices with prescribed similarity classes. Linear Multilinear Algebra. 1988; 24: 51-58.
10.1080/03081088808817897
Google Scholar
3Thompson R. Interlacing inequalities for invariant factors. Linear Algebra Appl. 1979; 24: 1-31.
10.1016/0024-3795(79)90144-7
Web of Science® Google Scholar
4Zaballa I. Pole assignment and additive perturbations of fixed rank. SIAM J Matrix Anal Appl. 1991; 12(1): 16-23.
10.1137/0612003
Web of Science® Google Scholar
5De Terán F, Dopico F. Low rank perturbation of Kronecker structures without full rank. SIAM J Matrix Anal Appl. 2007; 29(2): 496-529.
10.1137/060659922
Web of Science® Google Scholar
6De Terán F, Dopico F, Moro J. Low rank perturbation of Weierstrass structure. SIAM J Matrix AnalAppl. 2008; 30(2): 538-547.
10.1137/050633020
Web of Science® Google Scholar
7Gernandt H, Trunk C. Eigenvalue placement for regular matrix pencils with rank one perturbations. SIAM J Matrix Anal Appl. 2017; 38(1): 134-154.
10.1137/16M1066877
Web of Science® Google Scholar
8Baragaña I, Dodig M, Roca A, Stošić M. Bounded rank perturbations of regular pencils over arbitrary fields. Linear Algebra Appl. 2020; 601: 180-188.
10.1016/j.laa.2020.05.015
Web of Science® Google Scholar
9Baragaña I, Roca A. Rank-one perturbations of matrix pencils. Linear Algebra Appl. 2020; 606: 170-191.
10.1016/j.laa.2020.07.030
Web of Science® Google Scholar
10Dodig M, Stoşić M. Rank one perturbations of matrix pencils. SIAM J Matrix Anal Appl. 2020; 41(4): 1889-1911.
10.1137/19M1279411
Web of Science® Google Scholar
11den Boer H, Thijsse G. Semi-stability of sums of partial multiplicities under additive perturbation. Integr Equ Oper Theory. 1980; 3(1): 23-42.
10.1007/BF01682870
Google Scholar
12Markus AS, Parilis EÈ. The change of the Jordan structure of a matrix under small perturbations. Linear Algebra Appl. 1983; 51: 139-152.
10.1016/0024-3795(83)90210-0
Web of Science® Google Scholar
13Gracia J, de Hoyos I, Zaballa I. Perturbation of Linear Control Systems. Linear Algebra Appl. 1989; 121: 353-383.
10.1016/0024-3795(89)90710-6
Web of Science® Google Scholar
14de Hoyos I. Perturbación de matrices rectangulares y haces de matrices. PhD thesis. Universidad del País Vasco, Euskal Heriko Unibertsitatea, Bilbao; 1990.
Google Scholar
15Gracia J, de Hoyos I. Safety neighbourhoods for the kronecker canonical form; 1998. http://www.vc.ehu.es/campus/centros/farmacia/deptos-f/depme/gracia1.htm
Google Scholar
16Beitia M, de Hoyos I, Zaballa I. The change of the Jordan structure under one row perturbations. Linear Algebra Appl. 2005; 401: 119-134.
10.1016/j.laa.2004.02.003
Web of Science® Google Scholar
17Beitia M, de Hoyos I, Zaballa I. The change of similarity invariants under row perturbations: generic cases. Linear Algebra Appl. 2008; 409: 482-496.
10.1016/j.laa.2008.03.006
Web of Science® Google Scholar
18Beitia M, de Hoyos I, Zaballa I. The change of similarity invariants under row perturbations. Linear Algebra Appl. 2008; 409: 1302-1333.
10.1016/j.laa.2008.02.022
Web of Science® Google Scholar
19Dodig M, Stošić M. The change of feedback invariants under one row perturbation. Linear Algebra Appl. 2007; 422: 582-603.
10.1016/j.laa.2006.11.017
Web of Science® Google Scholar
20Zaballa I. Matrices with prescribed rows and invariant factors. Linear Algebra Appl. 1987; 87: 113-146.
10.1016/0024-3795(87)90162-5
Web of Science® Google Scholar
21Dodig M. Feedback invariants of matrices with prescribed rows. Linear Algebra Appl. 2005; 405: 121-154.
10.1016/j.laa.2005.03.002
Web of Science® Google Scholar
22Dodig M. Matrix pencils completion problems. Linear Algebra Appl. 2008; 428: 259-304.
10.1016/j.laa.2007.08.029
Web of Science® Google Scholar
23Dodig M. Explicit solution of the row completion problem for matrix pencils. Linear Algebra Appl. 2010; 432: 1299-1309.
10.1016/j.laa.2009.10.039
Web of Science® Google Scholar
24Dodig M. Completion up to a matrix pencil with column minimal indices as the only nontrivial Kronecker invariants. Linear Algebra Appl. 2013; 438: 3155-3173.
10.1016/j.laa.2012.12.040
Web of Science® Google Scholar
25Marshall W, Olkin I. Inequalities: Theory of Majorization and Its Applications. Academic Press; 1979.
Google Scholar
26Dodig M, Stošić M. On convexity of polynomial paths and generalized majorizations. Electron J Comb. 2010; 17(1): 61.
10.37236/333
Web of Science® Google Scholar
27Gantmacher F. Matrix Theory. Vol I, II. Chelsea Publishing; 1974.
Google Scholar
28Friedland S. Matrices: Algebra, Analysis and Applications. World Scientific; 2016.
Google Scholar
29Roca A. Asignación de Invariantes en Sistemas de Control. PhD thesis. Universitat Politècnica València; 2003.
Google Scholar
30Bathia R. Matrix Analysis. Springer; 1997.
Google Scholar
31Marques de Sà E. Imbedding conditions for $λ$ -matrices. Linear Algebra Appl. 1979; 24: 33-50.
10.1016/0024-3795(79)90145-9
Web of Science® Google Scholar
32Kalogeropoulos G, Psarrakos P, Karcanias N. On the computation of the Jordan canonical form of regular matrix polynomials. Linear Algebra Appl. 2004; 385: 117-130.
10.1016/j.laa.2003.06.017
Web of Science® Google Scholar
33Lippert R, Strang G. The Jordan forms of $AB$ and $BA$ . Electron J Int Linear Algebra Soc. 2009; 18: 281-288.
10.13001/1081-3810.1313
Web of Science® Google Scholar

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The change of the Weierstrass structure under one row perturbation

Abstract

1 INTRODUCTION

2 NOTATION AND BASIC DEFINITIONS

3 PREVIOUS RESULTS

3.1 Perturbation results

3.2 Completion results

4 MAIN RESULTS

5 CONCLUSIONS AND FUTURE RESEARCH

ACKNOWLEDGMENT

CONFLICT OF INTEREST

Biographies

REFERENCES

References

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The change of the Weierstrass structure under one row perturbation

Abstract

1 INTRODUCTION

2 NOTATION AND BASIC DEFINITIONS

3 PREVIOUS RESULTS

3.1 Perturbation results

3.2 Completion results

4 MAIN RESULTS

5 CONCLUSIONS AND FUTURE RESEARCH

ACKNOWLEDGMENT

CONFLICT OF INTEREST

Biographies

REFERENCES

References

Related

Information