UNITARILY INVARIANT NORM AND Q-NORM ESTIMATIONS FOR THE MOORE–PENROSE INVERSE OF MULTIPLICATIVE PERTURBATIONS OF MATRICES

Let B be a multiplicative perturbation of A ∈ Cm×n given by B = D∗ 1AD2, where D1 ∈ Cm×m and D2 ∈ Cn×n are both nonsingular. New upper bounds for ∥B† −A∥U and ∥B† −A∥Q are derived, where A†, B† are the Moore-Penrose inverses of A and B, and ∥ · ∥U , ∥ · ∥Q are any unitarily invariant norm and Q-norm, respectively. Numerical examples are provided to illustrate the sharpness of the obtained upper bounds.


Introduction
Throughout this paper, N, R, R + , C, C n and C m×n are the sets of positive integers, real numbers, nonnegative real numbers, complex numbers, column vector of ndimensions and m × n complex matrices, respectively. Let 0 m×n be the zero matrix of C m×n . When m = n, let I m denote the identity matrix of C m×m . An element P ∈ C m×m is said to be an orthogonal projection if P 2 = P and P * = P .
For any A ∈ C m×n , let R(A), N (A), A T , A * , ∥A∥ F , ∥A∥ 2 , ∥A∥ U and ∥A∥ Q denote the range, the null space, the transpose, the conjugate transpose, the Frobenius norm, the 2-norm, any unitarily invariant norm and any Q-norm of A, respectively. The Moore-Penrose inverse of A [6], written A † , is the unique element of C n×m which satisfies The Moore-Penrose inverse has various applications. One research field of the Moore-Penrose inverse is its perturbation theory. In this paper, we deal with norm estimations for the Moore-Penrose inverse associated with rank-preserving perturbations of matrices. Let A ∈ C m×n be given and B ∈ C m×n be a perturbation of A. Clearly, rank(B) = rank(A) if and only if there exist D 1 ∈ C m×m and D 2 ∈ C n×n such that B = D * 1 AD 2 , where D 1 and D 2 are both nonsingular. (1.1) The matrix B given by (1.1) is usually called a multiplicative perturbation of A, and norm estimations for ∥B † − A † ∥ U and ∥B † − A † ∥ Q are studied in the literatures [2,5,8], where ∥ · ∥ U and ∥ · ∥ Q denote any unitarily invariant norm and Q-norm [1], respectively. Upper bounds for ∥B † − A † ∥ U and ∥B † − A † ∥ Q are figured out in [2,Theorems 4.1 and 4.2] firstly. The results obtained in [2] are improved in [8, Theorems 3.1, 3.3 and 3.5], which are improved further in [5,Theorems 3.1 and 3.2]. Note that the Frobenius norm and the 2-norm are two special kinds of Q-norms. In a recent paper [4], new upper bounds for ∥B † − A † ∥ F and ∥B † − A † ∥ 2 are derived without using the Singular Value Decomposition (SVD), which serves however as the main tool in [2,5,8]. Based on the new method employed in [4], improvements of [5, Theorems 2.1, 3.1 and 3.2] are made in the special cases of the Frobenius norm and the 2-norm.
The purpose of this paper is to generalize the main results of [4] from the Frobenius norm and the 2-norm to the general unitarily invariant norm and Q-norm. Let B be a multiplicative perturbation of A ∈ C m×n given by (1.1). In this paper, we focus on the study of norm estimations for ∥B † − A † ∥ U and ∥B † − A † ∥ Q , and have managed to derive new upper bounds along the line initiated in [4, Theorems 2.2 and 3.3]. Thus, the main results of [5] are improved in the cases of the unitarily invariant norm and the Q-norm; see the comparison of (3.30) with (3.37), and (4.3) with (4.4).
The rest of this paper is organized as follows. In Section 2, we put forward some basic knowledge about the unitarily invariant norm, especially a norm equality of P −P Q and Q−QP is provided in the case that dim R(P )=dim R(Q), where P and Q are orthogonal projections acting on the same finite-dimensional Hilbert space. Let B be a multiplicative perturbation of A ∈ C m×n given by (1.1), and ∥ · ∥ U and ∥ · ∥ Q be any unitarily invariant norm and Q-norm. In Sections 3 and 4, we focus on the study of upper bounds for ∥B † − A † ∥ U and ∥B † − A † ∥ Q , respectively. Finally in Section 5, we provide three numerical examples to illustrate the sharpness of the upper bounds (3.30) and (4.3).

Some properties of the unitarily invariant norm
The term of the unitarily invariant norm can be found in [1, Sec. IV.2], which is originally defined on C n×n for some n ∈ N. We extend such a term in two steps. An extension to C m×n , called the (m, n)-unitarily invariant norm, is given in C m×n is given in Definition 2.2, which is the exact meaning of the unitarily invariant norm adopted in this paper. The purpose of this section is to put forward some basic knowledge about this new kind of unitarily invariant norm. Now, we give the definitions as follows: Definition 2.1. Let m, n ∈ N be given. An (m, n)-unitarily invariant norm is a norm ∥ · ∥ defined on the linear space C m×n such that ∥A∥ = ∥U AV ∥, for any A ∈ C m×n and any unitary matrices U ∈ C m×m and V ∈ C n×n . such that for each m, n ∈ N, the restriction of ∥ · ∥ U to C m×n is an (m, n)-unitarily invariant norm, and for any A ∈ C m×n , any k, l ∈ N, it holds that Trivial as it is, Lemma 2.1 below is stated for the sake of completeness.
Lemma 2.1. Let ∥ · ∥ U be any unitarily invariant norm. Then for any A ∈ C m×n , it holds that ∥A∥ U = ∥A * ∥ U .
Recall that any (n, n)-unitarily invariant norm is a symmetric norm, which can be stated as follows: An extension of the preceding lemma is as follows: Corollary 2.1. Let ∥ · ∥ U be any unitarily invariant norm. Then for any A ∈ C m×n , B ∈ C n×k and C ∈ C k×l , it holds that Similarly, define B and C. Then hence by Lemma 2.2 we have At the end of this section, we state a result of [7] as follows.

Unitarily invariant norm estimations for the Mo ore-Penrose inverse of multiplicative perturbations of matrices
Throughout this section, ∥ · ∥ U is any unitarily invariant norm. Let B be a multiplicative perturbation of A ∈ C m×n given by ( Proof. Since both D 1 and D 2 are nonsingular, we have The conclusion follows immediately from Theorem 2.1. Theorem 3.1. Let B be a multiplicative perturbation of A ∈ C m×n given by (1.1). Then for any s 1 , s 2 , s 3 , s 4 , s 5 ∈ C, we have where ) ) Therefore, we have First, we derive an upper bound for ∥Ω 1 ∥ U . Since B = D * 1 AD 2 , we have (3.9) The first equation above yields It follows from (3.5) and (3.10) that for any s 1 ∈ C, where The equations above, together with Corollary 2.1 and Lemma 2.1 yield Next, we derive upper bounds for ∥Ω 2 ∥ U and ∥Ω 3 ∥ U . By (3.6)-(3.7), Corollary 2.1 and Lemmas 2.1 and 3.1, we have

15)
It follows from (3.9) that for any s 2 , s 3 , s 4 , s 5 ∈ C, Taking * -operation we get It follows from (3.15) and (3.18) that Similarly, by (3.14) and (3.19) we can get We may combine (3.16), (3.17), (3.20) with (3.21) to get 1 * BD −1 2 , which means that A is also a multiplicative perturbation of B. In view of such an observation, by the preceding theorem we get the following corollary:

Corollary 3.1. Let B be a multiplicative perturbation of A ∈ C m×n given by (1.1).
Then for any u 1 , u 2 , u 3 , u 4 , u 5 ∈ C, we have

Numerical examples
In Note that each ∥ · ∥ p is a unitarily invariant norm, which is furthermore a Q-norm if p ≥ 2 [1, p95].   The details are listed in Table 2, where ∆ 1 and ∆ 2 are the relative errors of upper bound (4.3) and upper bound (4.4), respectively.