Fischer Type Log-Majorization of Singular Values on Partitioned Positive Semidefinite Matrices

We denote by Mn,m the vector space of all complex n ×m matrices. In particular, when n =m, Mn represents the set of square matrices of order n. Let A and B be Hermitian matrices. We use the notation A ≤ B or B ≥ A to mean that B − A is positive semidefinite. Particularly, B ≥ 0 (B > 0) means that B is positive semidefinite (B is positive definite). The singular values of A ∈Mn are the nonnegetive square roots of the eigenvalues of A∗A. We denote that sjðAÞ is the jth largest singular value of A ∈Mn and we denote sðAÞ = ðs1ðAÞ, s2ðAÞ,⋯, snðAÞÞ. Denote by ∥·∥∞ the spectral norm. For A ∈Mn, it is evident that ∥A∥∞ = s1ðAÞ. We rearrange the components of x = ðx1, x2,⋯, xnÞ ∈Rn in decreasing order as x1⁄21 ≥ x1⁄22 ≥⋯≥ x1⁄2n . Let x = ðx1,⋯, xnÞ, y = ðy1,⋯, ynÞ ∈Rn. If


Introduction
We denote by M n,m the vector space of all complex n × m matrices. In particular, when n = m, M n represents the set of square matrices of order n. Let A and B be Hermitian matrices. We use the notation A ≤ B or B ≥ A to mean that B − A is positive semidefinite. Particularly, B ≥ 0 (B > 0) means that B is positive semidefinite (B is positive definite).
The singular values of A ∈ M n are the nonnegetive square roots of the eigenvalues of A * A. We denote that s j ðAÞ is the jth largest singular value of A ∈ M n and we denote sðAÞ = ðs 1 ðAÞ, s 2 ðAÞ, ⋯, s n ðAÞÞ. Denote by ∥·∥ ∞ the spectral norm. For A ∈ M n , it is evident that ∥A∥ ∞ = s 1 ðAÞ.
We rearrange the components of x = ðx 1 , x 2 , ⋯, x n Þ ∈ ℝ n in decreasing order as x ½1 ≥ x ½2 ≥ ⋯ ≥ x ½n . Let x = ðx 1 , ⋯, x n Þ, y = ðy 1 , ⋯, y n Þ ∈ ℝ n . If then we say that x is weakly majorized by y and denotes x≺ ω y. If x≺ ω y and ∑ n i=1 x i = ∑ n i=1 y i , then we say that x is majorized by y and denotes x ≺ y. Let nonnegative vectors then we say that x is weakly log-majorized by y and denotes x≺ ω log y. If x≺ ω log y and Q n i=1 x i = Q n i=1 y i , then we say that x is log-majorized by y and denotes x≺ log y.
A norm on M n is called unitarily invariant if ∥UAV∥ = ∥A∥ for any A ∈ M n and any unitary U, V ∈ M n . For A ∈ M n and 1 ≤ k ≤ n, the kth compound matrix of A is denoted by C k ðAÞ. We list one of the useful properties on compound matrices: for A, B ∈ M n , C k ðABÞ = C k ðAÞC k ðBÞ. A series of properties of compound matrices can be seen in [1][2][3]. A complex matrix C is called contraction if C * C ≤ I or equivalently s 1 ðCÞ ≤ 1. We denote the block matrix by A ⊕ C. The well-known Fischer's inequality for determinant on the partitioned positive semidefinite matrix is the following. Let Then Let the eigenvalues of A ∈ M n be λ 1 , λ 2 , ⋯, λ n with |λ 1 | ≥ | λ 2 | ≥⋯≥|λ n |. Weyl [4] proved that f|λ i | g n i=1 ≺ log sðAÞ. Since for positive semidefinite matrices, singular values and eigenvalues are the same. Fischer's inequality can be rewritten as the following.
The motivation of this paper is to give the logmajorization relationship between the singular values of partitioned positive semidefinite matrix and its main diagonal matrix, which generalizes classical Fischer's inequality. In addition, we will also establish some singular value inequalities between partitioned positive semidefinite matrix and its main diagonal matrix.

Main Results
First, we list some lemmas that are used in our proofs.
where s i is the ith largest singular value of W.

Proof. W is a contractive matrix if and only if
For positive semidefinite matrices, singular values and eigenvalues are the same. Note that by Lemma 3 and spectral mapping theorem, this completes the proof.?
Next, we will establish the log-majorization relationship between the singular values of partitioned positive semidefinite matrix and its main diagonal matrix.
be positive semidefinite matrix with A ∈ M p , C ∈ M q ðp ≥ qÞ.
Then, there exists a contractive matrix W ∈ M p,q such that Proof. Applying Lemma 2, there exists a contractive matrix W such that B = A 1/2 WC 1/2 . Then, 2

Journal of Function Spaces
Denote The equality (16) can be rewritten as H = P * QP: Using compound matrices and the fact that for any square matrix In addition, for k = p + q, we have where the last equality is due to Lemma 4, this completes the proof.?

Corollary 6. Let
be positive semidefinite matrix with A ∈ M p , C ∈ M q ðp ≤ qÞ. Then, there exists a contractive matrix W ∈ M p,q such that Remark 7. From the equality (21), we can see that det ðHÞ ≤ det ðA ⊕ CÞ. The equality holds if and only if sðWÞ = 0, i.e., W = 0; it also implies that B = 0.

Corollary 8. Let
be positive semidefinite matrix with A ∈ M p , C ∈ M q . Then, there exists a contractive matrix W ∈ M p,q such that for all unitarily invariant norms k·k.
Proof. Recall that weak log-majorization is stronger than weak majorization, i.e., x≺ ω log y implies x≺ ω y. By Theorem 5, we have It is clear that Therefore, we have Using the Fan Dominance Principle (see [6]), this completes the proof.?
In the following section, we will establish some singular value inequalities between partitioned positive semidefinite matrix and its main diagonal matrix. The following wellknown result is due to Ky Fan [1].

Journal of Function Spaces
In particular, Theorem 10. Let be positive semidefinite matrix with A ∈ M p , C ∈ M q ðp ≥ qÞ.
Then, there exists a contractive matrix W ∈ M p,q such that In addition, for 1 ≤ i ≤ q, Proof.
In (33), change the position of and By similar proof method, we can get the following inequality.
Using Lemma 4, we can divide into the following cases: This completes the proof.?
The following theorem is proved by using the result given by Hirzallah and Kittaneh [7].
Remark 13. One may ask whether the inequality holds But it needs not be true. For example, setting Then,

Data Availability
The data used to support the findings of this study are included within the article.

Conflicts of Interest
The author declares that he have no conflicts of interest.