Relative and Absolute Perturbation Bounds for Weighted Polar Decomposition

Let Cm×n, Cm×n r , C m ≥ , C m > , and In denote the set of m × n complex matrices, subset of Cm×n consisting of matrices with rank r, set of the Hermitian nonnegative definite matrices of order m, subset of C ≥ 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 M ∈ C > ,N ∈ C >. For A ∈ Cm×n, we denote by R A , r A , A∗, AMN N−1A∗M,AMN, ‖A‖ 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 AMN and A † MN can be found in details in 1, 2 . The weighted polar decomposition MN-WPD of A ∈ Cm×n is given by


Introduction
Let C m×n , C m×n r , C m ≥ , C m > , 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 C m ≥ 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 M ∈ C m > , N ∈ C n > .For A ∈ C m×n , we denote by R A , r A , A * , A # MN N −1 A * M, A † MN , A 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 A # MN and A † MN can be found in details in 1, 2 .The weighted polar decomposition MN-WPD of A ∈ C m×n is given by where Q is an M, N weighted partial isometric matrix 3, 4 and H satisfies NH ∈ C n ≥ .In this case, Q and H are called the M, N weighted unitary polar factor and generalized nonnegative polar factor, respectively, of this decomposition.

Journal of Applied Mathematics
Yang and Li 5 proved that the MN-WPD is unique under the condition 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 A have the same rank. Let have the generalized polar decompositions A QH and A Q H.For the perturbation bound of the subunitary unitary polar factors, the following two results can be found in 16

1.4
For the nonnegative polar factors, the perturbation bounds obtained by Chen 17 are 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.

Notation and Some Lemmas
Firstly, we introduce the definitions of the weighted norms.
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 A ∈ C m×n r and A ∈ C m×n s have their weighted singular value decompositions MN-SVDs :

2.2
Then the MN-WPDs of A QH and A Q H can be obtained by . ., σ r and Σ 1 diag σ 1 , σ 2 , . . ., σ s .Here The following three lemmas can be found from 22 , 23 and 16 , respectively.Lemma 2.2.Let B 1 and B 2 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 B 1 and the other contains that of B 2 .If η > 0, then there exists a unique solution X to the matrix equation B 1 X − XB 2 P and, moreover, Lemma 2.3.Let Ω ∈ C s×s and Γ ∈ C t×t be two Hermitian matrices, and let and, moreover, where Lemma 2.4.Let S S 1 , S 2 ∈ C m×m and T T 1 , T 2 ∈ C n×n be both unitary matrices, where S 1 ∈ C m×r , T 1 ∈ C n×s .Then for any matrix B ∈ C m×n , one has 2.6

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.
, and let A QH and A Q H be their MN-WPDs of A and A, respectively.Then Proof.By 2.1 , and 2.2 the perturbation E can be written as which together with the facts that Taking the conjugate transpose on both sides of 3.4 and subtracting it from 3.3 leads to Applying Lemma 2.2 to 3.7 for the Frobenius norm leads to Since it follows from Definition 2.1 and the fact that

3.13
which proves the theorem.

3.14
Proof.From the MN-SVDs of A and A in 2.1 and 2.2 and the facts that , the weighted Moore-Penrose inverses of A and A can be written as that is, 3.17 By 3.17 , we can obtain

3.18
Similarly, by 19 respectively.By the first equations in 3.18 -3.21 , we derive

3.23
Applying Lemma 2.3 to 3.22 and 3.23 , respectively, and noting that η min 1≤i≤s,1≤j≤r we find that

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.
Proof.By 2.1 , 2.2 , and 2.3 , we have Let ΔH H − H, we rewrite 4.3 Premultiplying and postmultiplying both sides of 4.5 by, respectively, Similarly, we have Applying Lemma 2.2 to 4.6 gives

4.8
Notice that Combining 4.7 -4.9 gives which proves the theorem.
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.
Proof.Applying Lemma 2.3 to 4.6 gives

4.15
From 16 , we know max 1≤i≤r 1≤j≤s 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 4.9.Let A, A A E ∈ C m×n n , and let A QH and A Q H be their MN-WPDs of A and A, respectively.Then The relative perturbation bound for the generalized nonnegative polar factors is given in the following theorem.
which together with Lemma 2.2 gives

4.25
By 4.7 and the facts that

4.26
It follows from 4.9 , 4.25 and 4.26 that

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.
0 are the nonzero M, N weighted singular values of A and A, respectively.

Remark 3 . 2 . 3 . 3 . 3 .
If M I m and N I n in Theorem 3.1, the bound 3.1 is reduced to bound 1.Theorem Let A ∈ C m×n r and A A E ∈ C m×n s , and let A QH and A Q H be their MN-WPDs of A and A, respectively.Then

,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 .

Theorem 4 . 1 .
Let A ∈ C m×n r and A A E ∈ C m×n s , and let A QH and A Q H be their MN-WPDs of A and A, respectively.Then

Remark 4 . 2 .Corollary 4 . 3 .Corollary 4 . 4 . 12 Corollary 4 . 5 .
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.Let A ∈ C m×n n and A A E ∈ C m×n s , and let A QH and A Q H be their MN-WPDs of A and A, respectively.Then Let A ∈ C m×n r and A A E ∈ C m×n n , and let A QH and A Q H be their MN-WPDs of A and A, respectively.ThenH − H NN ≤ σ 1 σ 1 σ r σ n 1 E MN .4.Let A, A A E ∈ C m×n n, and let A QH and A Q H be their MN-WPDs of A and A, respectively.Then

Theorem 4 . 6 .
Let A ∈ C m×n r and A A E ∈ C m×n s , and let A QH and A Q H be their MN-WPDs of A and A, respectively.Then

Corollary 4 . 7 .
Let A ∈ C m×n n and A A E ∈ C m×n s , and let A QH and A Q H be their MN-WPDs of A and A, respectively.Then

Corollary 4 . 8 .
Let A ∈ C m×n r and A A E ∈ C m×n n , and let A QH and A Q H be their MN-WPDs of A and A, respectively.Then

Theorem 4 . 10 .
Let A ∈ C m×n r and A A E ∈ C m×n s , and let A QH and A Q H be their MN-WPDs of A and A, respectively.Then 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., and let A QH and A Q H be their MN-WPDs of A and A, respectively.Then