A collection of efficient tools to work with almost strictly sign regular matrices

Definition 1.A real matrix $A={(a_{ij})}_{1\le i,j,\le n}$ is called type-I staircase matrix if it satisfies simultaneously the following conditions:

• (a)

  $a_{ii}\ne 0$, $\forall i\in \{1,\dots ,n\}$;

• (b)

  $a_{ij}=0,i>j$ $\Rightarrow a_{kl}=0$, if $l\le j$, $i\le k$;

• (c)

  $a_{ij}=0,i<j$ $\Rightarrow a_{kl}=0$, if $k\le i$, $j\le l$.

Definition 2.A matrix A is called type-II staircase if $P_{n}A$ is a type-I staircase matrix.

Example 1.Given the matrix A, we generate the matrix $B_2$ with ones above the main diagonal and in the main diagonal and bellow it the element is 1 if $a_{ij}\ne 0$ and 0 if it is zero.

Then we apply the function to $B_2$.

and we can see in the matrix A the meaning of this vector.

Example 2.Given A the matrix of the Example 1, the output of the command

Example 3.Given A the matrix of the Example 1, the output of the command

Now, we highlight some entries of the matrix of the Example 1 in two different ways

Example 4.Given A the matrix of the Example 1, the output of the command

Definition 3.Given a vector $\epsilon =({\epsilon }_1,{\epsilon }_2,\dots ,{\epsilon }_n)\in {ℝ}^n$, we say that $\epsilon$ is a signature sequence, or simply, is a signature, if ${\epsilon }_i\in \{+1,-1\}$ for all $i\le n$.

Definition 4.A real matrix $A={(a_{ij})}_{1\le i,j,\le n}$ is said to be SR, with signature $\epsilon =({\epsilon }_1,{\epsilon }_2,\dots ,{\epsilon }_n)$, if all its minors satisfy that

Definition 5.Let $A={\left(a_{ij}\right)}_{1\le i,j\le n}$ be a type-I (type-II) staircase matrix. A submatrix $A[\alpha |\beta ]$, with $\alpha ,\beta \in Q_{m,n}$, is said to be nontrivial if all its main diagonal (secondary diagonal) elements are nonzero.

Definition 6. ([11])A real matrix $A={(a_{ij})}_{1\le i,j\le n}$ type-I or type-II staircase is said to be ASSR, with signature $\epsilon =({\epsilon }_1,{\epsilon }_2,\dots ,{\epsilon }_n)$, if all its nontrivial minors $\det A[\alpha |\beta ]$ satisfy that

Remark 1.Observe that an ASSR matrix is SR, since the trivial minors are zero and the nontrivial minors satisfy the strict inequality (9). Observe also that an ASSR matrix is nonsingular.

Theorem 1.Let A be a real $n\times n$ matrix and $\epsilon =({\epsilon }_1,{\epsilon }_2,\dots ,{\epsilon }_n)$ be a signature. Then A is ASSR with signature $\epsilon$ if and only if A is a type-I or type-II staircase matrix, and all its nontrivial minors with $\alpha ,\beta \in Q_{m,n}^0$, $m\le n$, satisfy

Theorem 2.Let $B={(b_{ij})}_{1\le i,j\le n}$ be a nonsingular type-I staircase matrix, with zero pattern defined by I, J, $\widehat{I}$, and $\widehat{J}$. If B is ASSR with signature $\epsilon =({\epsilon }_1,{\epsilon }_2,\dots ,{\epsilon }_n)$, then the NE of B can be performed without row exchanges and the pivots $p_{ij}$ satisfy, for $1\le j\le i\le n$,

where ${\epsilon }_0=1$ and $j_t$ is defined in (7).

Theorem 3.Let $B={(b_{ij})}_{1\le i,j\le n}$ be a nonsingular type-II staircase matrix, with zero pattern defined by I, J, $\widehat{I}$, and $\widehat{J}$. If B is ASSR with signature $\epsilon =({\epsilon }_1,{\epsilon }_2,\dots ,{\epsilon }_n)$, then the NE of $B^T$ can be performed without row exchanges and the pivots $q_{ij}$ satisfy, for $1\le i<j\le n$,

where ${\epsilon }_0=1$ and ${\widehat{i}}_t$ is defined in (8).

Remark 2.Using this theorems it is easy to verify that ${\epsilon }_{j-j_t}{\epsilon }_{j-j_t+1}{p}_{ij}={\epsilon }_{j-1}{\epsilon }_j{p}_{ij}$ and ${\epsilon }_{i-{\widehat{i}}_t}{\epsilon }_{i-{\widehat{i}}_t+1}{q}_{ij}={\epsilon }_{i-1}{\epsilon }_i{q}_{ij}$

Remark 3.Let A be a type-I (type-II) staircase matrix. Then, in relation to $A_h$ ($A_h^T$) matrices, the next results are verified:

1. $A_h$ and $A_h^T$ are also type-I (type-II) staircase matrix for all $h\in \{1,2,\dots ,n\}$.
2. If the zero pattern of A is given by $I=\{i_0,\dots ,i_{\ell }\}$, $J=\{j_0,\dots ,j_{\ell }\}$, $\widehat{I}=\{{î}_0,\dots ,{î}_r\}$, $\widehat{J}=\{{ĵ}_0,\dots ,{ĵ}_r\}$, then the zero pattern of $A_h$ matrices is given by $I^h=\{1,i_a-h+1,\dots ,i_l-h+1\}$, $J^h=\{1,j_a-h+1,\dots ,j_l-h+1\}$, ${\widehat{I}}^h=\{1,{î}_b-h+1,\dots ,{î}_r-h+1\}$, ${\widehat{J}}^h=\{1,{ĵ}_b-h+1,\dots ,{ĵ}_r-h+1\}$, where $a=\min \{s/j_s-h+1\ge 2\}$ and $b=\min \{s/{î}_s-h+1\ge 2\}$.
3. If A is ASSR matrix with signature $\epsilon =({\epsilon }_1,{\epsilon }_2,\dots ,{\epsilon }_n)$ then $A_h$ and $A_h^T$ are ASSR matrices with signature ${\epsilon }^{\prime }=({\epsilon }_1,{\epsilon }_2,\dots ,{\epsilon }_{n-h+1})$.

Theorem 4.A nonsingular matrix $A={\left(a_{ij}\right)}_{1\le i,j\le n}$ is ASSR with signature $\epsilon =({\epsilon }_1,{\epsilon }_2,\dots ,$ ${\epsilon }_n)$, with ${\epsilon }_2=1$ if and only if for every $h=1,\dots ,n-1$ the following properties hold simultaneously:

• a)

  A is type-I staircase;

• b)

  the NE of the matrices $A_h=A[h,\dots ,n]$ and $A_h^T$ can be performed without row exchanges;

• c)

  the pivots $p_{ij}^h$ of the NE of $A_h$ satisfy conditions corresponding to (12), (13), and the pivots $q_{ij}^h$ of the NE $A_h^T$ satisfy (14) and (15);

• d)

  for the positions $(i^h,j^h)$ of matrix $A_h$:

  • if $i^h\ge j^h$ and $i^h-j^h=i_t^h-j_t^h$ then ${\epsilon }_{j^h-j_t^h}{\epsilon }_{j^h-j_t^h+1}={\epsilon }_{j^h-1}{\epsilon }_{j^h}$,

  • if $i^h<j^h$ and $i^h-j^h={\widehat{i}}_t^h-{\widehat{j}}_t^h$ then ${\epsilon }_{i^h-{\widehat{i}}_t^h}{\epsilon }_{i^h-{\widehat{i}}_t^h+1}={\epsilon }_{i^h-1}{\epsilon }_{i^h}$,

  where indices $i_t^h,j_t^h,{\widehat{i}}_t^h,{\widehat{j}}_t^h$ are given by conditions corresponding to (7) and (8).

Example 5.Given the matrices

If we apply the function Signature we obtain that M is ASSR and N is not.

Example 6.Let A be the matrix

By applying function Neville to $A_1$ and $A_1^t$, the pivots $p_{ij}$ and $q_{ij}$ (we represent all in one matrix ${\bar{P}}_1$ where ${\bar{p}}_{ij}=p_{ij}$ if $i\ge j$ and ${\bar{p}}_{ij}=q_{ij}$ if $i\le j$) and the signs of the pivots are, Attending to the signs of the pivots, a candidate to be the signature is ${\epsilon }^1=(1,-1,-1,1)$.

Now, if we apply the function Neville to $A_2$ and $A_2^t$, the pivots $p_{ij}$ and $q_{ij}$ and the signs of the pivots are,

Attending to the signs of the pivots, a candidate to be the signature is ${\epsilon }^{(2)}=(1,1,-1)$ which must be coherent with ${\epsilon }^{(1)}$, that is, must be equal to the first three elements of ${\epsilon }^{(1)}$, but it is not ${\epsilon }_2^{(1)}=-1\ne 1={\epsilon }_2^{(2)}$. Clearly, this matrix is not ASSR because the following minors of order 2 have different signs: