Some new perturbation bounds for both weighted unitary polar factors and generalized nonnegative polar factors of the weighted polar decompositions are presented without the restriction that A and its perturbed matrix have the same rank. These bounds improve the corresponding recent results.

1. Introduction

Let

, and I_n denote the set of m × n complex matrices, subset of C^m×n consisting of matrices with rank r, set of the Hermitian nonnegative definite matrices of order m, subset of

consisting of positive-definite matrices and n × n unit matrix, respectively. Without specification, we always assume that m > n > max {r, s} and the given weight matrices

. For A ∈ C^m×n, we denote by

and ∥A∥_F the column space, rank, conjugate transpose, weighted conjugate transpose (or adjoint), weighted Moore-Penrose inverse, unitarily invariant norm, and Frobenius norm of A, respectively. The definitions of

and

can be found in details in [1, 2]. The weighted polar decomposition (MN-WPD) of A ∈ C^m×n is given by

(1.1)

where Q is an (M, N) weighted partial isometric matrix [3, 4] and H satisfies

. In this case, Q and H are called the (M, N) weighted unitary polar factor and generalized nonnegative polar factor, respectively, of this decomposition.

Yang and Li [5] proved that the MN-WPD is unique under the condition

(1.2)

In this paper, we always assume that the MN-WPD satisfies condition (1.2).

If M = I_m and N = I_n, then the MN-WPD is reduced to the generalized polar decomposition and Q and H are reduced to the subunitary polar factor and nonnegative polar factor, respectively. Further, if r(A) = n, then the MN-WPD is just the polar decomposition and Q and H are just the unitary polar factor and positive polar factor.

The problem on estimating the perturbation bounds for both polar decomposition and generalized polar decomposition under the assumption that the matrix and its perturbed matrix have the same rank [6–15] attracted most attention, and only some attention was given without the restriction [16, 17]. However, the arbitrary perturbation case seems important in both theoretical and practical problems. Now we list some published bounds for (generalized) polar decomposition without the restriction that A and have the same rank.

Let

have the (generalized) polar decompositions A = QH and

. For the perturbation bound of the (subunitary) unitary polar factors, the following two results can be found in [16]

(1.3)

(1.4)

For the nonnegative polar factors, the perturbation bounds obtained by Chen [17] are

(1.5)

(1.6)

It is known that different elements of a vector are usually needed to be given some different weights in practice (e.g., the residual of the linear system), and the problems with weights, such as weighted generalized inverses problem and weighted least square problem, draw more and more attention, see, for example, [1, 2, 18, 19]. As a generalization of the (generalized) polar decomposition, MN-WPD may be useful for these problems. Therefore, it is of interest to study MN-WPD and its related properties.

Our goal of this paper is mainly to generalize the perturbation bounds in (1.3)–(1.6) to those for the weighted polar factors of the MN-WPDs in the corresponding weighted norms. The rest of this paper is organized as follows.

In Section 2, we list notation and some lemmas which are useful in the sequel. In Section 3, we present an absolute perturbation bound and a relative perturbation bound for the weighted unitary polar factors, respectively, and some perturbation bounds for the generalized nonnegative polar factors are also given in Section 4.

2. Notation and Some Lemmas

Firstly, we introduce the definitions of the weighted norms.

Definition 2.1. Let A ∈ C^m×n. The norms ∥A∥_(MN) = ∥M^1/2AN^−1/2∥ and ∥A∥_F(MN) = ∥M^1/2AN^−1/2∥_F are called the weighted unitarily invariant norm and weighted Frobenius norm of A, respectively. The definitions of ∥A∥_(MN) and ∥A∥_F(MN) can be also found in [20, 21].

Let and have their weighted singular value decompositions (MN-SVDs):

(2.1)

(2.2)

Then the MN-WPDs of A = QH and

can be obtained by

(2.3)

where

and

satisfy

and

, and

and

. Here σ₁ ≥ σ₂ ≥ ⋯≥σ_r > 0 and

are the nonzero (M, N) weighted singular values of A and

, respectively.

The following three lemmas can be found from [22], [23] and [16], respectively.

Lemma 2.2. Let B₁ and B₂ be two Hermitian matrices and let P be a complex matrix. Suppose that there are two disjoint intervals separated by a gap of width at least η, where one interval contains the spectrum of B₁ and the other contains that of B₂. If η > 0, then there exists a unique solution X to the matrix equation B₁X − XB₂ = P and, moreover,

(2.4)

Lemma 2.3. Let Ω ∈ C^s×s and Γ ∈ C^t×t be two Hermitian matrices, and let E, F ∈ C^s×t. If λ(Ω)∩λ(Γ) = ⌀, then ΩX − XΓ = ΩE + FΓ has a unique solution X ∈ C^s×t, and, moreover,

(2.5)

where

Lemma 2.4. Let S = (S₁, S₂) ∈ C^m×m and T = (T₁, T₂) ∈ C^n×n be both unitary matrices, where S₁ ∈ C^m×r, T₁ ∈ C^n×s. Then for any matrix B ∈ C^m×n, one has

(2.6)

3. Perturbation Bounds for the Weighted Unitary Polar Factors

In this section, we present an absolute perturbation bound and a relative perturbation bound for the weighted unitary polar factors.

Theorem 3.1. Let and , and let A = QH and be their MN-WPDs of A and , respectively. Then

(3.1)

Proof. By (2.1), and (2.2) the perturbation E can be written as

(3.2)

which together with the facts that

and

gives

(3.3)

(3.4)

(3.5)

(3.6)

Taking the conjugate transpose on both sides of (3.4) and subtracting it from (3.3) leads to

(3.7)

Applying Lemma 2.2 to (3.7) for the Frobenius norm leads to

(3.8)

Since

(3.9)

it follows from Definition 2.1 and the fact that U^*M^1/2,

, N^−1/2V, and

are all unitary matrices that

(3.10)

(3.11)

Adding (3.10) to (3.11) gives

(3.12)

Combing (3.5), (3.6), (3.8), (3.12), Lemma 2.4, and the fact that

gets

(3.13)

which proves the theorem.

Remark 3.2. If M = I_m and N = I_n in Theorem 3.1, the bound (3.1) is reduced to bound (1.3).

Theorem 3.3. Let and , and let A = QH and be their MN-WPDs of A and , respectively. Then

(3.14)

Proof. From the MN-SVDs of A and in (2.1) and (2.2) and the facts that and , the weighted Moore-Penrose inverses of A and can be written as

(3.15)

Premultiplying the equation

leads to

(3.16)

that is,

(3.17)

By (3.17), we can obtain

(3.18)

Similarly, by

and

, we get

(3.19)

(3.20)

(3.21)

respectively. By the first equations in (3.18)–(3.21), we derive

(3.22)

(3.23)

Applying Lemma 2.3 to (3.22) and (3.23), respectively, and noting that

(3.24)

we find that

(3.25)

(3.26)

From (3.12), the second equations in (3.18)–(3.21), (3.25), (3.26), and Lemma 2.4, we deduce that

(3.27)

which proves the theorem.

Remark 3.4. If M = I_m and N = I_n in Theorem 3.3, the bound (3.14) is reduced to bound (1.4).

4. Perturbation Bounds for the Generalized Nonnegative Polar Factors

In this section, two absolute perturbation bounds and a relative perturbation bound for the generalized nonnegative polar factors are given.

Theorem 4.1. Let and , and let A = QH and be their MN-WPDs of A and , respectively. Then

(4.1)

Proof. By (2.1), (2.2), and (2.3), we have

(4.2)

which give

(4.3)

Let

, we rewrite (4.3)

(4.4)

that is,

(4.5)

Premultiplying and postmultiplying both sides of (4.5) by, respectively,

and N⁻¹V₁ give

(4.6)

Similarly, we have

(4.7)

Applying Lemma 2.2 to (4.6) gives

(4.8)

Notice that

(4.9)

Combining (4.7)–(4.9) gives

(4.10)

which proves the theorem.

Remark 4.2. If M = I_m and N = I_n in Theorem 4.1, the bound (4.1) is reduced to bound (1.5).

If r = n, s < n or s = n, r < n or r = s = n, we can easily derive the following three corollaries.

Corollary 4.3. Let and , and let A = QH and be their MN-WPDs of A and , respectively. Then

(4.11)

Corollary 4.4. Let and , and let A = QH and be their MN-WPDs of A and , respectively. Then

(4.12)

Corollary 4.5. Let , and let A = QH and be their MN-WPDs of A and , respectively. Then

(4.13)

If we take the weighted Frobenius norm as the specific weighted unitarily invariant norm in Theorem 4.1, an alternative absolute perturbation bound can be derived as follows.

Theorem 4.6. Let and , and let A = QH and be their MN-WPDs of A and , respectively. Then

(4.14)

Proof. Applying Lemma 2.3 to (4.6) gives

(4.15)

From [16], we know

(4.16)

which together with (4.7), (4.9), and (4.15) gives

(4.17)

Hence, we complete the theorem.

Similarly, we can obtain the following three corollaries.

Corollary 4.7. Let and , and let A = QH and be their MN-WPDs of A and , respectively. Then

(4.18)

Corollary 4.8. Let and , and let A = QH and be their MN-WPDs of A and , respectively. Then

(4.19)

Corollary 4.9. Let , and let A = QH and be their MN-WPDs of A and , respectively. Then

(4.20)

The relative perturbation bound for the generalized nonnegative polar factors is given in the following theorem.

Theorem 4.10. Let and , and let A = QH and be their MN-WPDs of A and , respectively. Then

(4.21)

Proof. From the proof of Theorem 3.3, we know that

(4.22)

Premultiplying and postmultiplying both sides of (4.5) by, respectively,

and

give

(4.23)

Premultiplying and postmultiplying both sides of (4.23) by, respectively,

and U₁ give

(4.24)

which together with Lemma 2.2 gives

(4.25)

By (4.7) and the facts that

and

, we have

(4.26)

It follows from (4.9), (4.25) and (4.26) that

(4.27)

which proves the theorem.

Remark 4.11. If M = I_m and N = I_n in Theorem 4.10, the bound (4.21) is reduced to bound (1.6).

The following three corollaries can be also easily obtained.

Corollary 4.12. Let and , and let A = QH and be their MN-WPDs of A and , respectively. Then

(4.28)

Corollary 4.13. Let and , and let A = QH and be their MN-WPDs of A and , respectively. Then

(4.29)

Corollary 4.14. Let and , and let A = QH and be their MN-WPDs of A and , respectively. Then

(4.30)

5. Conclusion

In this paper, we obtain the relative and absolute perturbation bounds for the weighted polar decomposition without the restriction that the original matrix and its perturbed matrix have the same rank. These bounds are the corresponding generalizations of those for the (generalized) polar decomposition.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China (no. 11171361) and in part by the Natural Science Foundation Project of CQ CSTC (2010BB9215).

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Relative and Absolute Perturbation Bounds for Weighted Polar Decomposition

Abstract

1. Introduction

2. Notation and Some Lemmas

3. Perturbation Bounds for the Weighted Unitary Polar Factors

4. Perturbation Bounds for the Generalized Nonnegative Polar Factors

5. Conclusion

Acknowledgments

References

References

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Relative and Absolute Perturbation Bounds for Weighted Polar Decomposition

Abstract

1. Introduction

2. Notation and Some Lemmas

3. Perturbation Bounds for the Weighted Unitary Polar Factors

4. Perturbation Bounds for the Generalized Nonnegative Polar Factors

5. Conclusion

Acknowledgments

References

References

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