Relative and Absolute Perturbation Bounds for Weighted Polar Decomposition

Abstract

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 ^0𝑥0𝐴^02𝑑𝑐 have the same rank. These bounds improve the corresponding recent results.

1. Introduction

Let 𝐶^𝑚×𝑛,𝐶_𝑟^𝑚×𝑛,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⁻¹𝐴^∗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 by𝐴=𝑄𝐻, (1.1) 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.

Yang and Li [5] proved that the MN-WPD is unique under the condition𝑅𝑄^#_𝑀𝑁=𝑅(𝐻). (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 𝐴 have the same rank.

Let 𝐴∈𝐶_𝑟^𝑚×𝑛,𝐴=𝐴+𝐸∈𝐶_𝑠^𝑚×𝑛 have the (generalized) polar decompositions 𝐴=𝑄𝐻 and 𝐴=𝑄𝐻. For the perturbation bound of the (subunitary) unitary polar factors, the following two results can be found in [16]𝑄−𝑄_𝐹≤1min𝜎_𝑟,𝜎_𝑠‖𝐸‖_𝐹,𝑄−𝑄_𝐹≤√22𝐴^†𝐸²_𝐹+𝐸𝐴^†2_𝐹+𝐴^†𝐸²_𝐹+𝐸𝐴^†2_𝐹. (1.3)(1.4)

For the nonnegative polar factors, the perturbation bounds obtained by Chen [17] are≤𝜎𝐻−𝐻¹+𝜎₁𝜎_𝑟+𝜎_𝑠‖𝐸‖,≤𝜎+2𝐻−𝐻¹𝜎₁𝜎₁+𝜎₁𝐸𝐴^†+𝐸𝐴^†+𝜎₁𝐴^†𝐸+𝜎₁𝐴^†𝐸. (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 𝐴∈𝐶_𝑟^𝑚×𝑛 and 𝐴∈𝐶_𝑠^𝑚×𝑛 have their weighted singular value decompositions (MN-SVDs): 𝐴=𝑈Σ𝑉^∗=𝑈₁,𝑈₂Σ₁0𝑉00¹,𝑉₂^∗=𝑈₁Σ₁𝑉^∗₁𝑈Σ𝑉,𝐴=^∗=𝑈₁,𝑈₂Σ₁0𝑉00¹,𝑉₂^∗=𝑈₁Σ₁𝑉^∗₁. (2.1)(2.2) Then the MN-WPDs of 𝐴=𝑄𝐻 and 𝐴=𝑄𝐻 can be obtained by 𝑄=𝑈₁𝑉^∗₁,𝐻=𝑁⁻¹𝑉₁Σ₁𝑉^∗₁,𝑄=𝑈₁𝑉^∗₁,𝐻=𝑁⁻¹𝑉₁Σ₁𝑉^∗₁, (2.3) where 𝑈=(𝑈₁,𝑈₂),𝑈=(𝑈₁,𝑈₂)∈𝐶^𝑚×𝑚 and 𝑉=(𝑉₁,𝑉₂),𝑉=(𝑉₁,𝑉₂)∈𝐶^𝑛×𝑛 satisfy 𝑈^∗𝑀𝑈=𝑈^∗𝑀𝑈=𝐼_𝑚 and 𝑉^∗𝑁⁻¹𝑉=𝑉^∗𝑁⁻¹V=𝐼_𝑛, and 𝑈₁∈𝐶_𝑟^𝑚×𝑟,𝑈₁∈𝐶_𝑠^𝑚×𝑠,𝑉₁∈𝐶_𝑟^𝑛×𝑟,𝑉₁∈𝐶_𝑠^𝑛×𝑠,Σ₁=diag(𝜎₁,𝜎₂,…,𝜎_𝑟) and Σ₁=diag(𝜎₁,𝜎₂,…,𝜎_𝑠). Here 𝜎₁≥𝜎₂≥⋯≥𝜎_𝑟>0 and 𝜎₁≥𝜎₂≥⋯≥𝜎_𝑠>0 are the nonzero (𝑀,𝑁) weighted singular values of 𝐴 and 𝐴, 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, ‖𝑋‖≤1𝜂‖𝑃‖. (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)=Ø, then Ω𝑋−𝑋Γ=Ω𝐸+𝐹Γ has a unique solution 𝑋∈𝐶^𝑠×𝑡, and, moreover, ‖𝑋‖_𝐹≤1𝜂‖𝐸‖²_𝐹+‖𝐹‖²_𝐹, (2.5) where 𝜂=min_{𝜆∈𝜆(Ω),}𝜆_∈𝜆(Γ)(|𝜆−𝜆|/|𝜆|²+|𝜆|²).

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 ‖𝐵‖²_𝐹=𝑆^∗₁𝐵𝑇₁²_𝐹+𝑆^∗₁𝐵𝑇₂²_𝐹+𝑆^∗₂𝐵𝑇₂²_𝐹+𝑆^∗₂𝐵𝑇₂²_𝐹. (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 𝐴∈𝐶_𝑟^𝑚×𝑛 and 𝐴=𝐴+𝐸∈𝐶_𝑠^𝑚×𝑛, and let 𝐴=𝑄𝐻 and 𝐴=𝑄𝐻 be their MN-WPDs of 𝐴 and 𝐴, respectively. Then 𝑄−𝑄_{𝐹(𝑀𝑁)}≤1min𝜎_𝑟,𝜎_𝑠‖𝐸‖_{𝐹(𝑀𝑁)}. (3.1)

Proof. By (2.1), and (2.2) the perturbation 𝐸 can be written as 𝐸=𝐴−𝐴=𝑈₁Σ₁𝑉^∗₁−𝑈₁Σ₁𝑉^∗₁, (3.2) which together with the facts that 𝑈^∗₁𝑀𝑈₁=𝑉^∗₁𝑁⁻¹𝑉₁=𝐼_𝑟 and 𝑈^∗₁𝑀𝑈₁=𝑉^∗₁𝑁⁻¹𝑉₁=𝐼_𝑠 gives 𝑈^∗₁𝑀𝐸𝑁⁻¹𝑉₁=𝑈^∗₁𝑀𝑈₁Σ₁−Σ₁𝑉^∗₁𝑁⁻¹𝑉₁,𝑈^∗₁𝑀𝐸𝑁⁻¹𝑉₁=Σ₁𝑉^∗₁𝑁⁻¹𝑉₁−𝑈^∗₁𝑀𝑈₁Σ₁𝑈_∗2,𝑀𝐸𝑁⁻¹𝑉₁=−𝑈^∗₂𝑀𝑈₁Σ₁,𝑈^∗₂𝑀𝐸𝑁⁻¹𝑉₁=𝑈^∗₂𝑀𝑈₁Σ₁𝑈_∗1,𝑀𝐸𝑁⁻¹𝑉₂=Σ₁𝑉^∗₁𝑁⁻¹𝑉₂,𝑈^∗₁𝑀𝐸𝑁⁻¹𝑉₂=−Σ₁𝑉^∗₁𝑁⁻¹𝑉₂. (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 Σ₁𝑈^∗₁𝑀𝑈₁−𝑉^∗₁𝑁⁻¹𝑉₁+𝑈^∗₁𝑀𝑈₁−𝑉^∗₁𝑁⁻¹𝑉₁Σ₁=𝑈^∗₁𝑀𝐸𝑁⁻¹𝑉₁−𝑉^∗₁𝑁⁻¹𝐸^∗𝑀𝑈₁. (3.7) Applying Lemma 2.2 to (3.7) for the Frobenius norm leads to 𝑈^∗₁𝑀𝑈₁−𝑉^∗₁𝑁⁻¹𝑉_1𝐹≤1𝜎_𝑟+𝜎_𝑠𝑈^∗₁𝑀𝐸𝑁⁻¹𝑉_1𝐹+𝑉^∗₁𝑁⁻¹𝐸^∗𝑀𝑈_1𝐹. (3.8) Since 𝑈^∗𝑀𝑄−𝑄𝑁⁻¹𝑈𝑉=^∗1𝑀𝑈₁−𝑉^∗₁𝑁⁻¹𝑉₁−𝑉^∗₁𝑁⁻¹𝑉₂𝑈^∗₂𝑀𝑈₁0,𝑈^∗𝑀𝑄−𝑄𝑁⁻¹𝑉𝑉=^∗1𝑁⁻¹𝑉₁−𝑈^∗₁𝑀𝑈₁𝑉^∗₁𝑁⁻¹𝑉₂−𝑈^∗₂𝑀𝑈₁0, (3.9) it follows from Definition 2.1 and the fact that 𝑈^∗𝑀^1/2, 𝑈^∗𝑀^1/2, 𝑁^−1/2𝑉, and 𝑁^−1/2𝑉 are all unitary matrices that 𝑄−𝑄²_{𝐹(𝑀𝑁)}=𝑈^∗𝑀𝑄−𝑄𝑁⁻¹𝑉²_𝐹=𝑈^∗₁𝑀𝑈₁−𝑉^∗₁𝑁⁻¹𝑉₁²_𝐹+𝑉^∗₁𝑁⁻¹𝑉₂²_𝐹+𝑈^∗₂𝑀𝑈₁²_𝐹,𝑄−𝑄²_{𝐹(𝑀𝑁)}=𝑈^∗𝑀𝑄−𝑄𝑁⁻¹𝑉²_𝐹=𝑉^∗₁𝑁⁻¹𝑉₁−𝑈^∗₁𝑀𝑈₁²_𝐹+𝑉^∗₁𝑁⁻¹𝑉₂²_𝐹+𝑈^∗₂𝑀𝑈₁²_𝐹. (3.10)(3.11) Adding (3.10) to (3.11) gives 2𝑄−𝑄²_{𝐹(𝑀𝑁)}=𝑈^∗₁𝑀𝑈₁−𝑉^∗₁𝑁⁻¹𝑉₁²_𝐹+𝑉^∗₁𝑁⁻¹𝑉₂²_𝐹+𝑈^∗₂𝑀𝑈₁²_𝐹+𝑉^∗₁𝑁⁻¹𝑉₁−𝑈^∗₁𝑀𝑈₁²_𝐹+𝑉^∗₁𝑁⁻¹𝑉₂²_𝐹+𝑈^∗₂𝑀𝑈₁²_𝐹=2𝑈^∗₁𝑀𝑈₁−𝑉^∗₁𝑁⁻¹𝑉₁²_𝐹+𝑉^∗₁𝑁⁻¹𝑉₂²_𝐹+𝑉^∗₁𝑁⁻¹𝑉₂²_𝐹+𝑈^∗₂𝑀𝑈₁²_𝐹+𝑈^∗₂𝑀𝑈₁²_𝐹. (3.12) Combing (3.5), (3.6), (3.8), (3.12), Lemma 2.4, and the fact that ‖𝑈^∗₁𝑀𝑈₁−𝑉^∗₁𝑁⁻¹𝑉₁‖²_𝐹=‖𝑉^∗₁𝑁⁻¹𝑉₁−𝑈^∗₁𝑀𝑈₁‖²_𝐹 gets 2𝑄−𝑄²_{𝐹(𝑀𝑁)}1≤2𝜎_𝑟+𝜎_𝑠²𝑈^∗₁𝑀𝐸𝑁⁻¹𝑉_1𝐹+𝑉^∗₁𝑁⁻¹𝐸^∗𝑀𝑈_1𝐹²+1𝜎²_𝑟𝑈^∗₁𝑀𝐸𝑁⁻¹𝑉₂²_𝐹+1𝜎²_𝑠𝑈^∗₂𝑀𝐸𝑁⁻¹𝑉₁²_𝐹+1𝜎²_𝑠𝑈^∗₁𝑀𝐸𝑁⁻¹𝑉₂²_𝐹+1𝜎²_𝑟𝑈^∗₂𝑀𝐸𝑁⁻¹𝑉₁²_𝐹≤2𝜎_𝑟+𝜎_𝑠²𝑈^∗₁𝑀𝐸𝑁⁻¹𝑉₁²_𝐹+𝑉^∗₁𝑁⁻¹𝐸^∗𝑀𝑈₁²_𝐹+1min𝜎²_𝑟,𝜎²_𝑠𝑈^∗₁𝑀𝐸𝑁⁻¹𝑉₂²_𝐹+𝑈^∗₂𝑀𝐸𝑁⁻¹𝑉₁²_𝐹+𝑈^∗₁𝑀𝐸𝑁⁻¹𝑉₂²_𝐹+𝑈^∗₂𝑀𝐸𝑁⁻¹𝑉₁²_𝐹≤1min𝜎²_𝑟,𝜎²_𝑠𝑈^∗₁𝑀𝐸𝑁⁻¹𝑉₁²_𝐹+𝑈^∗₁𝑀𝐸𝑁⁻¹𝑉₂²_𝐹+𝑈^∗₂𝑀𝐸𝑁⁻¹𝑉₁²_𝐹+𝑈^∗₁𝑀𝐸𝑁⁻¹𝑉₁²_𝐹+𝑈^∗₁𝑀𝐸𝑁⁻¹𝑉₂²_𝐹+𝑈^∗₂𝑀𝐸𝑁⁻¹𝑉₁²_𝐹≤2min𝜎²_𝑟,𝜎²_𝑠𝑀^1/2𝐸𝑁^−1/22_𝐹=2min𝜎²_𝑟,𝜎²_𝑠‖𝐸‖²_{𝐹()𝑀𝑁}, (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 𝐴∈𝐶_𝑟^𝑚×𝑛 and 𝐴=𝐴+𝐸∈𝐶_𝑠^𝑚×𝑛, and let 𝐴=𝑄𝐻 and 𝐴=𝑄𝐻 be their MN-WPDs of 𝐴 and 𝐴, respectively. Then 𝑄−𝑄_{𝐹(𝑀𝑁)}≤√22𝐴^†_𝑀𝑁𝐸²_{𝐹()𝑁𝑁}+𝐴^†_𝑀𝑁𝐸²_{𝐹()𝑁𝑁}+𝐸𝐴^†_𝑀𝑁²_{𝐹()𝑀𝑀}+𝐸𝐴^†_𝑀𝑁²_{𝐹()𝑀𝑀}. (3.14)

Proof. From the MN-SVDs of 𝐴 and 𝐴 in (2.1) and (2.2) and the facts that 𝑈^∗𝑀𝑈=𝑈^∗𝑀𝑈=𝐼_𝑚 and 𝑉^∗𝑁⁻¹𝑉=𝑉^∗𝑁⁻¹𝑉=𝐼_𝑛, the weighted Moore-Penrose inverses of 𝐴 and 𝐴 can be written as 𝐴^†_𝑀𝑁=𝑁⁻¹𝑉₁Σ₁⁻¹𝑈^∗₁𝑀,𝐴^†_𝑀𝑁=𝑁⁻¹𝑉₁Σ₁⁻¹𝑈^∗₁𝑀. (3.15) Premultiplying the equation 𝐴−𝐴=𝐸 by 𝐴^†_𝑀𝑁 leads to 𝐴^†_𝑀𝑁𝐴−𝐴^†_𝑀𝑁𝐴=𝐴^†_𝑀𝑁𝐸, (3.16) that is, 𝑁⁻¹𝑉₁Σ₁⁻¹𝑈^∗₁𝑀𝑈₁Σ₁𝑉^∗₁−𝑁⁻¹𝑉₁𝑉^∗₁=𝐴^†_𝑀𝑁𝐸. (3.17) By (3.17), we can obtain 𝑈^∗₁𝑀𝑈₁Σ₁−Σ₁𝑉^∗₁𝑁⁻¹𝑉₁=Σ₁𝑉^∗₁𝐴^†_𝑀𝑁𝐸𝑁⁻¹𝑉₁,−𝑉^∗₁𝑁⁻¹𝑉₂=𝑉^∗₁𝐴^†_𝑀𝑁𝐸𝑁⁻¹𝑉₂. (3.18) Similarly, by 𝐴^†_𝑀𝑁𝐴−𝐴^†_𝑀𝑁𝐴=𝐴^†_𝑀𝑁𝐸, 𝐴𝐴^†_𝑀𝑁−𝐴𝐴^†_𝑀𝑁=𝐸𝐴^†_𝑀𝑁 and 𝐴𝐴^†_𝑀𝑁−𝐴𝐴^†_𝑀𝑁=𝐸𝐴^†_𝑀𝑁, we get Σ₁𝑉^∗₁𝑁⁻¹𝑉₁−𝑈^∗₁𝑀𝑈₁Σ₁=Σ₁𝑉^∗₁𝐴^†_𝑀𝑁𝐸𝑁⁻¹𝑉₁,𝑉^∗₁𝑁⁻¹𝑉₂=𝑉^∗₁𝐴^†_𝑀𝑁𝐸𝑁⁻¹𝑉₂,Σ₁𝑉^∗₁𝑁⁻¹𝑉₁−𝑈^∗₁𝑀𝑈₁Σ₁=𝑈^∗₁𝑀𝐸𝐴^†_𝑀𝑁𝑈₁Σ₁,−𝑈^∗₂𝑀𝑈₁=𝑈^∗₂𝑀𝐸𝐴^†_𝑀𝑁𝑈₁𝑈_∗1,𝑀𝑈₁Σ₁−Σ₁𝑉^∗₁𝑁⁻¹𝑉₁=𝑈^∗₁𝑀𝐸𝐴^†_𝑀𝑁𝑈₁Σ₁,𝑈^∗₂𝑀𝑈₁=𝑈^∗₂𝑀𝐸𝐴^†_𝑀𝑁𝑈₁, (3.19)(3.20)(3.21) respectively. By the first equations in (3.18)–(3.21), we derive 𝑈^∗₁𝑀𝑈₁−𝑉^∗₁𝑁⁻¹𝑉₁Σ₁+Σ₁𝑈^∗₁𝑀𝑈₁−𝑉^∗₁𝑁⁻¹𝑉₁=Σ₁𝑉^∗₁𝐴^†_𝑀𝑁𝐸𝑁⁻¹𝑉₁−𝑉^∗₁𝐴^†_𝑀𝑁𝐸𝑁⁻¹𝑉₁^∗Σ₁,𝑈^∗₁𝑀𝑈₁−𝑉^∗₁𝑁⁻¹𝑉₁Σ₁+Σ₁𝑈^∗₁𝑀𝑈₁−𝑉^∗₁𝑁⁻¹𝑉₁=𝑈^∗₁𝑀𝐸𝐴^†_𝑀𝑁𝑈₁Σ₁−Σ₁𝑈^∗₁𝑀𝐸𝐴^†_𝑀𝑁𝑈₁^∗. (3.22)(3.23) Applying Lemma 2.3 to (3.22) and (3.23), respectively, and noting that 𝜂=min_{1≤𝑖≤𝑠,1≤𝑗≤𝑟}𝜎_𝑖+𝜎_𝑗𝜎²_𝑖+𝜎²_𝑗≥1, (3.24) we find that 𝑈^∗₁𝑀𝑈₁−𝑉^∗₁𝑁⁻¹𝑉₁²_𝐹≤𝑉^∗₁𝐴^†_𝑀𝑁𝐸𝑁⁻¹𝑉₁²_𝐹+𝑉^∗₁𝐴^†_𝑀𝑁𝐸𝑁⁻¹𝑉₁²_𝐹,𝑈^∗₁𝑀𝑈₁−𝑉^∗₁𝑁⁻¹𝑉₁²_𝐹≤𝑈^∗₁𝑀𝐸𝐴^†_𝑀𝑁𝑈₁²_𝐹+𝑈^∗₁𝑀𝐸𝐴^†_𝑀𝑁𝑈₁²_𝐹. (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 2𝑄−𝑄²_{𝐹(𝑀𝑁)}≤𝑉^∗₁𝐴^†_𝑀𝑁𝐸𝑁⁻¹𝑉₁²_𝐹+𝑉^∗₁𝐴^†_𝑀𝑁𝐸𝑁⁻¹𝑉₁²_𝐹+𝑈^∗₁𝑀𝐸𝐴^†_𝑀𝑁𝑈₁²_𝐹+𝑈^∗₁𝑀𝐸𝐴^†_𝑀𝑁𝑈₁²_𝐹+𝑉^∗₁𝐴^†_𝑀𝑁𝐸𝑁⁻¹𝑉₂²_𝐹+𝑈^∗₂𝑀𝐸𝐴^†_𝑀𝑁𝑈₁²_𝐹+𝑉^∗₁𝐴^†_𝑀𝑁𝐸𝑁⁻¹𝑉₂²_𝐹+𝑈^∗₂𝑀𝐸𝐴^†_𝑀𝑁𝑈₁²_𝐹=𝑉^∗₁𝐴^†_𝑀𝑁𝐸𝑁⁻¹𝑉₁²_𝐹+𝑉^∗₁𝐴^†_𝑀𝑁𝐸𝑁⁻¹𝑉₂²_𝐹+𝑉^∗₁𝐴^†_𝑀𝑁𝐸𝑁⁻¹𝑉₁²_𝐹+𝑉^∗₁𝐴^†_𝑀𝑁𝐸𝑁⁻¹𝑉₂²_𝐹+𝑈^∗₁𝑀𝐸𝐴^†_𝑀𝑁𝑈₁²_𝐹+𝑈^∗₂𝑀𝐸𝐴^†_𝑀𝑁𝑈₁²_𝐹+𝑈^∗₁𝑀𝐸𝐴^†_𝑀𝑁𝑈₁²_𝐹+𝑈^∗₂𝑀𝐸𝐴^†_𝑀𝑁𝑈₁²_𝐹≤𝑁^1/2𝐴^†_𝑀𝑁𝐸𝑁^−1/22_𝐹+𝑁^1/2𝐴^†_𝑀N𝐸𝑁^−1/22_𝐹+𝑀^1/2𝐸𝐴^†_𝑀𝑁𝑀^−1/22_𝐹+𝑀^1/2𝐸𝐴^†_𝑀𝑁𝑀^−1/22_𝐹=𝐴^†_𝑀𝑁𝐸²_{𝐹(𝑁𝑁)}+𝐴^†_𝑀𝑁𝐸²_{𝐹(𝑁𝑁)}+𝐸𝐴^†_𝑀𝑁²_{𝐹(𝑀𝑀)}+𝐸𝐴^†_𝑀𝑁²_{𝐹(𝑀𝑀)}, (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 𝐴∈𝐶_𝑟^𝑚×𝑛 and 𝐴=𝐴+𝐸∈𝐶_𝑠^𝑚×𝑛, and let 𝐴=𝑄𝐻 and 𝐴=𝑄𝐻 be their MN-WPDs of 𝐴 and 𝐴, respectively. Then 𝐻−𝐻_(𝑁𝑁)≤𝜎₁+𝜎₁𝜎_𝑟+𝜎_𝑠+2‖𝐸‖_(𝑀𝑁). (4.1)

Proof. By (2.1), (2.2), and (2.3), we have 𝑁𝐻²=𝐴^∗𝑀𝐴,𝑁𝐻²=𝐴^∗𝑀𝐴, (4.2) which give 𝑁𝐻𝐻−𝐻+𝑁𝐻−𝐻𝐻=𝐴^∗𝑀𝐴−𝐴+𝐴−𝐴^∗𝑀𝐴. (4.3) Let Δ𝐻=𝐻−H, we rewrite (4.3) 𝑁𝐻Δ𝐻+𝑁Δ𝐻𝐻=𝐴^∗𝑀𝐸+𝐸^∗𝑀𝐴, (4.4) that is, 𝑉₁Σ₁𝑉^∗₁Δ𝐻+𝑁Δ𝐻𝑁⁻¹𝑉₁Σ₁𝑉^∗₁=𝑉₁Σ₁𝑈^∗₁𝑀𝐸+𝐸^∗𝑀𝑈₁Σ₁𝑉^∗₁. (4.5) Premultiplying and postmultiplying both sides of (4.5) by, respectively, 𝑉^∗₁𝑁⁻¹ and 𝑁⁻¹𝑉₁ give Σ₁𝑉^∗₁Δ𝐻𝑁⁻¹𝑉₁+𝑉^∗₁Δ𝐻𝑁⁻¹𝑉₁Σ₁=Σ₁𝑈^∗₁𝑀𝐸𝑁⁻¹𝑉₁+𝑉^∗₁𝑁⁻¹𝐸^∗𝑀𝑈₁Σ₁. (4.6) Similarly, we have 𝑉^∗₁Δ𝐻𝑁⁻¹𝑉₂=𝑈^∗₁𝑀𝐸𝑁⁻¹𝑉₂,𝑉^∗₂Δ𝐻𝑁⁻¹𝑉₁=𝑉^∗₂𝑁⁻¹𝐸^∗𝑀𝑈₁. (4.7) Applying Lemma 2.2 to (4.6) gives 𝑉^∗₁Δ𝐻𝑁⁻¹𝑉₁≤1𝜎_𝑟+𝜎_𝑠Σ₁𝑈^∗₁𝑀𝐸𝑁⁻¹𝑉₁+𝑉^∗₁𝑁⁻¹𝐸^∗𝑀𝑈₁Σ₁≤𝜎₁𝜎_𝑟+𝜎_𝑠𝑈^∗₁𝑀𝐸𝑁⁻¹𝑉₁+𝜎₁𝜎_𝑟+𝜎_𝑠𝑉^∗₁𝑁⁻¹𝐸^∗𝑀𝑈₁≤𝜎₁𝜎_𝑟+𝜎_𝑠‖𝐸‖_(𝑀𝑁)+𝜎₁𝜎_𝑟+𝜎_𝑠‖𝐸‖_(𝑀𝑁)=𝜎₁+𝜎₁𝜎_𝑟+𝜎_𝑠‖𝐸‖_(𝑀𝑁). (4.8) Notice that 𝑉^∗Δ𝐻𝑁⁻¹𝑉𝑉=^∗1Δ𝐻𝑁⁻¹𝑉₁𝑉^∗₁Δ𝐻𝑁⁻¹𝑉₂𝑉^∗₂Δ𝐻𝑁⁻¹𝑉₁0. (4.9) Combining (4.7)–(4.9) gives ‖Δ𝐻‖_(𝑁𝑁)=𝑉^∗Δ𝐻𝑁⁻¹𝑉≤𝑉^∗₁Δ𝐻𝑁⁻¹𝑉₁+𝑉^∗₁Δ𝐻𝑁⁻¹𝑉₂+𝑉^∗₂Δ𝐻𝑁⁻¹𝑉₁≤𝜎₁+𝜎₁𝜎_𝑟+𝜎_𝑠‖𝐸‖_(𝑀𝑁)+𝑈^∗₁𝑀𝐸𝑁⁻¹𝑉₂+𝑉^∗₂𝑁⁻¹𝐸^∗𝑀𝑈₁≤𝜎₁+𝜎₁𝜎_𝑟+𝜎_𝑠‖𝐸‖_(𝑀𝑁)+‖𝐸‖_(𝑀𝑁)+‖𝐸‖_(𝑀𝑁)=𝜎₁+𝜎₁𝜎_𝑟+𝜎_𝑠+2‖𝐸‖_(𝑀𝑁), (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 𝐴∈𝐶_𝑛^𝑚×𝑛 and 𝐴=𝐴+𝐸∈𝐶_𝑠^𝑚×𝑛, and let 𝐴=𝑄𝐻 and 𝐴=𝑄𝐻 be their MN-WPDs of 𝐴 and 𝐴, respectively. Then 𝐻−𝐻_(𝑁𝑁)≤𝜎₁+𝜎₁𝜎_𝑛+𝜎_𝑠+1‖𝐸‖_(𝑀𝑁). (4.11)

Corollary 4.4. Let 𝐴∈𝐶_𝑟^𝑚×𝑛 and 𝐴=𝐴+𝐸∈𝐶_𝑛^𝑚×𝑛, and let 𝐴=𝑄𝐻 and 𝐴=𝑄𝐻 be their MN-WPDs of 𝐴 and 𝐴, respectively. Then 𝐻−𝐻_(𝑁𝑁)≤𝜎₁+𝜎₁𝜎_𝑟+𝜎_𝑛+1‖𝐸‖_(𝑀𝑁). (4.12)

Corollary 4.5. Let 𝐴,𝐴=𝐴+𝐸∈𝐶_𝑛^𝑚×𝑛, and let 𝐴=𝑄𝐻 and 𝐴=𝑄𝐻 be their MN-WPDs of 𝐴 and 𝐴, respectively. Then 𝐻−𝐻_(𝑁𝑁)≤𝜎₁+𝜎₁𝜎_𝑛+𝜎_𝑛‖𝐸‖_(𝑀𝑁). (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 𝐴∈𝐶_𝑟^𝑚×𝑛 and 𝐴=𝐴+𝐸∈𝐶_𝑠^𝑚×𝑛, and let 𝐴=𝑄𝐻 and 𝐴=𝑄𝐻 be their MN-WPDs of 𝐴 and 𝐴, respectively. Then 𝐻−𝐻_{𝐹(𝑁𝑁)}≤2+√2max𝜎²₁+𝜎²_𝑠𝜎₁+𝜎_𝑠,𝜎²_𝑟+𝜎²₁𝜎_𝑟+𝜎₁‖𝐸‖_{𝐹(𝑀𝑁)}. (4.14)

Proof. Applying Lemma 2.3 to (4.6) gives 𝑉^∗₁Δ𝐻𝑁⁻¹𝑉₁²_𝐹≤max_{1≤𝑖≤𝑟1≤𝑗≤𝑠}𝜎²_𝑖+𝜎²_𝑗𝜎_𝑖+𝜎_𝑗²𝑈^∗₁𝑀𝐸𝑁⁻¹𝑉₁²_𝐹+𝑉^∗₁𝑁⁻¹𝐸^∗𝑀𝑈₁²_𝐹≤2max_{1≤𝑖≤𝑟1≤𝑗≤𝑠}𝜎²_𝑖+𝜎²_𝑗𝜎_𝑖+𝜎_𝑗²‖𝐸‖²_{𝐹(𝑀𝑁)}. (4.15) From [16], we know max_{1≤𝑖≤𝑟1≤𝑗≤𝑠}𝜎²_𝑖+𝜎²_𝑗𝜎_𝑖+𝜎_𝑗²𝜎=max²¹+𝜎²_𝑠𝜎₁+𝜎_𝑠²,𝜎²_𝑟+𝜎²₁𝜎_𝑟+𝜎₁², (4.16) which together with (4.7), (4.9), and (4.15) gives ‖Δ𝐻‖_{𝐹(𝑁𝑁)}=𝑉^∗Δ𝐻𝑁⁻¹𝑉_𝐹≤𝑉^∗₁Δ𝐻𝑁⁻¹𝑉_1𝐹+𝑉^∗₁Δ𝐻𝑁⁻¹𝑉_2𝐹+𝑉^∗₂Δ𝐻𝑁⁻¹𝑉_1𝐹≤√2max𝜎²₁+𝜎²_𝑠𝜎₁+𝜎_𝑠,𝜎²_𝑟+𝜎²₁𝜎_𝑟+𝜎₁‖𝐸‖_{𝐹(𝑀𝑁)}+𝑈^∗₁𝑀𝐸𝑁⁻¹𝑉_2𝐹+𝑉^∗₂𝑁⁻¹𝐸^∗𝑀𝑈_1𝐹≤√2max𝜎²₁+𝜎²_𝑠𝜎₁+𝜎_𝑠,𝜎²_𝑟+𝜎²₁𝜎_𝑟+𝜎₁‖𝐸‖_{𝐹(𝑀𝑁)}+‖𝐸‖_{𝐹(𝑀𝑁)}+‖𝐸‖_{𝐹(𝑀𝑁)}=√2max𝜎²₁+𝜎²_𝑠𝜎₁+𝜎_𝑠,𝜎²_𝑟+𝜎²₁𝜎_𝑟+𝜎₁+2‖𝐸‖_{𝐹(𝑀𝑁)}. (4.17) Hence, we complete the theorem.

Similarly, we can obtain the following three corollaries.

Corollary 4.7. Let 𝐴∈𝐶_𝑛^𝑚×𝑛 and 𝐴=𝐴+𝐸∈𝐶_𝑠^𝑚×𝑛, and let 𝐴=𝑄𝐻 and 𝐴=𝑄𝐻 be their MN-WPDs of 𝐴 and 𝐴, respectively. Then 𝐻−𝐻_{𝐹(𝑁𝑁)}≤1+√2max𝜎²₁+𝜎²_𝑠𝜎₁+𝜎_𝑠,𝜎²_𝑛+𝜎²₁𝜎_𝑛+𝜎₁‖𝐸‖_{𝐹(𝑀𝑁)}. (4.18)

Corollary 4.8. Let 𝐴∈𝐶_𝑟^𝑚×𝑛 and 𝐴=𝐴+𝐸∈𝐶_𝑛^𝑚×𝑛, and let 𝐴=𝑄𝐻 and 𝐴=𝑄𝐻 be their MN-WPDs of 𝐴 and 𝐴, respectively. Then 𝐻−𝐻_{𝐹(𝑁𝑁)}≤1+√2max𝜎²₁+𝜎²_𝑛𝜎₁+𝜎_𝑛,𝜎²_𝑟+𝜎²₁𝜎_𝑟+𝜎₁‖𝐸‖_{𝐹(𝑀𝑁)}. (4.19)

Corollary 4.9. Let 𝐴,𝐴=𝐴+𝐸∈𝐶_𝑛^𝑚×𝑛, and let 𝐴=𝑄𝐻 and 𝐴=𝑄𝐻 be their MN-WPDs of 𝐴 and 𝐴, respectively. Then 𝐻−𝐻_{𝐹(𝑁𝑁)}≤√2max𝜎²₁+𝜎²_𝑛𝜎₁+𝜎_𝑛,𝜎²_𝑛+𝜎²₁𝜎_𝑛+𝜎₁‖𝐸‖_{𝐹(𝑀𝑁)}. (4.20)

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

Theorem 4.10. Let 𝐴∈𝐶_𝑟^𝑚×𝑛 and 𝐴=𝐴+𝐸∈𝐶_𝑠^𝑚×𝑛, and let 𝐴=𝑄𝐻 and 𝐴=𝑄𝐻 be their MN-WPDs of 𝐴 and 𝐴, respectively. Then 𝐻−𝐻_(𝑁𝑁)≤𝜎₁𝜎₁𝜎₁+𝜎₁𝐸𝐴^†_{𝑀𝑁(𝑀𝑀)}+𝐸𝐴^†_{𝑀𝑁(𝑀𝑀)}+𝜎₁𝐴^†_𝑀𝑁𝐸_(𝑁𝑁)+𝜎₁𝐴^†_𝑀𝑁𝐸_(𝑁𝑁). (4.21)

Proof. From the proof of Theorem 3.3, we know that 𝐴^†_𝑀𝑁=𝑁⁻¹𝑉₁Σ₁⁻¹𝑈^∗₁𝑀,𝐴^†_𝑀𝑁=𝑁⁻¹𝑉₁Σ₁⁻¹𝑈^∗₁𝑀. (4.22) Premultiplying and postmultiplying both sides of (4.5) by, respectively, (𝐴^†_𝑀𝑁)^∗ and 𝐴^†_𝑀𝑁 give 𝑀𝑈₁𝑉^∗₁Δ𝐻𝑁⁻¹𝑉₁Σ₁⁻¹𝑈^∗₁𝑀+𝑀𝑈₁Σ₁⁻¹𝑉^∗₁Δ𝐻𝑁⁻¹𝑉₁𝑈^∗₁𝑀=𝑀𝑈₁𝑈^∗₁𝑀𝐸𝑁⁻¹𝑉₁Σ₁⁻¹𝑈^∗₁𝑀+𝑀𝑈₁Σ₁⁻¹𝑉^∗₁𝑁⁻¹𝐸^∗𝑀𝑈₁𝑈^∗₁𝑀=𝑀𝑈₁𝑈^∗₁𝑀𝐸𝐴^†_𝑀𝑁+𝐸𝐴^†_𝑀𝑁^∗𝑀𝑈₁𝑈^∗₁𝑀. (4.23) Premultiplying and postmultiplying both sides of (4.23) by, respectively, 𝑈^∗₁ and 𝑈₁ give 𝑉^∗₁Δ𝐻𝑁⁻¹𝑉₁Σ₁⁻¹+Σ₁⁻¹𝑉^∗₁Δ𝐻𝑁⁻¹𝑉₁=𝑈^∗₁𝑀𝐸𝐴^†_𝑀𝑁𝑈₁+𝑈^∗₁𝐸𝐴^†_𝑀𝑁^∗𝑀𝑈₁, (4.24) which together with Lemma 2.2 gives 𝑉^∗₁Δ𝐻𝑁⁻¹𝑉₁≤𝜎₁𝜎₁𝜎₁+𝜎₁𝑈^∗₁𝑀𝐸𝐴^†_𝑀𝑁𝑈₁+𝑈^∗₁𝐸𝐴^†_𝑀𝑁^∗𝑀𝑈₁≤𝜎₁𝜎₁𝜎₁+𝜎₁𝐸𝐴^†_{𝑀𝑁(𝑀𝑀)}+𝐸𝐴^†_{𝑀𝑁(𝑀𝑀)}. (4.25) By (4.7) and the facts that 𝐴𝐴^†_𝑀𝑁=𝑈₁𝑈^∗₁𝑀 and 𝐴𝐴^†_𝑀𝑁=𝑈₁𝑈^∗₁𝑀, we have 𝑉^∗₁Δ𝐻𝑁⁻¹𝑉₂=𝑈^∗₁𝑀𝐸𝑁⁻¹𝑉₂=𝑈^∗₁𝑀𝑈₁𝑈^∗₁𝑀𝐸𝑁⁻¹𝑉₂=𝑈^∗₁𝑀𝐴𝐴^†_𝑀𝑁𝐸𝑁⁻¹𝑉₂=Σ₁𝑉^∗₁𝐴^†_𝑀𝑁𝐸𝑁⁻¹𝑉₂,𝑉^∗₂Δ𝐻𝑁⁻¹𝑉₁=𝑉^∗₂𝑁⁻¹𝐸^∗𝑀𝑈₁=𝑉^∗₂𝑁⁻¹𝐸^∗𝑀𝑈₁𝑈^∗₁𝑀𝑈₁=𝑉^∗₂𝑁⁻¹𝐴^†_𝑀𝑁𝐸^∗𝐴^∗𝑀𝑈₁=𝑉^∗₂𝑁⁻¹𝐴^†_𝑀𝑁𝐸^∗𝑉₁Σ₁. (4.26) It follows from (4.9), (4.25) and (4.26) that ‖Δ𝐻‖_(𝑁𝑁)=𝑁^1/2Δ𝐻𝑁^−1/2=𝑉^∗Δ𝐻𝑁⁻¹𝑉≤𝑉^∗₁Δ𝐻𝑁⁻¹𝑉₁+𝑉^∗₁Δ𝐻𝑁⁻¹𝑉₂+𝑉^∗₂Δ𝐻𝑁⁻¹𝑉₁≤𝜎₁𝜎₁𝜎₁+𝜎₁𝐸𝐴^†_{𝑀𝑁(𝑀𝑀)}+𝐸𝐴^†_{𝑀𝑁(𝑀𝑀)}+Σ₁𝑉^∗₁𝐴^†_𝑀𝑁𝐸𝑁⁻¹𝑉₂+𝑉^∗₂𝑁⁻¹𝐴^†_𝑀𝑁𝐸^∗𝑉₁Σ₁≤𝜎₁𝜎₁𝜎₁+𝜎₁𝐸𝐴^†_{𝑀𝑁(𝑀𝑀)}+𝐸𝐴^†_{𝑀𝑁(𝑀𝑀)}+𝜎₁𝐴^†_𝑀𝑁𝐸_(𝑁𝑁)+𝜎₁𝐴^†_𝑀𝑁𝐸_(𝑁𝑁), (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 𝐴∈𝐶_𝑛^𝑚×𝑛 and 𝐴=𝐴+𝐸∈𝐶_𝑠^𝑚×𝑛, and let 𝐴=𝑄𝐻 and 𝐴=𝑄𝐻 be their MN-WPDs of 𝐴 and 𝐴, respectively. Then 𝐻−𝐻_(𝑁𝑁)≤𝜎₁𝜎₁𝜎₁+𝜎₁𝐸𝐴^†_{𝑀𝑁(𝑀𝑀)}+𝐸𝐴^†_{𝑀𝑁(𝑀𝑀)}+𝜎₁𝐴^†_𝑀𝑁𝐸_(𝑁𝑁). (4.28)

Corollary 4.13. Let 𝐴∈𝐶_𝑟^𝑚×𝑛 and 𝐴=𝐴+𝐸∈𝐶_𝑛^𝑚×𝑛, and let 𝐴=𝑄𝐻 and 𝐴=𝑄𝐻 be their MN-WPDs of 𝐴 and 𝐴, respectively. Then 𝐻−𝐻_(𝑁𝑁)≤𝜎₁𝜎₁𝜎₁+𝜎₁𝐸𝐴^†_{𝑀𝑁(𝑀𝑀)}+𝐸𝐴^†_{𝑀𝑁(𝑀𝑀)}+𝜎₁𝐴^†_𝑀𝑁𝐸_(𝑁𝑁). (4.29)

Corollary 4.14. Let 𝐴∈𝐶_𝑛^𝑚×𝑛 and 𝐴=𝐴+𝐸∈𝐶_𝑛^𝑚×𝑛, and let 𝐴=𝑄𝐻 and 𝐴=𝑄𝐻 be their MN-WPDs of 𝐴 and 𝐴, respectively. Then 𝐻−𝐻_(𝑁𝑁)≤𝜎₁𝜎₁𝜎₁+𝜎₁𝐸𝐴^†_{𝑀𝑁(𝑀𝑀)}+𝐸𝐴^†_{𝑀𝑁(𝑀𝑀)}. (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).