Singular Value and Matrix Norm Inequalities between Positive Semidefinite Matrices and Their Blocks

Abstract

In this paper, we obtain some inequalities involving positive semidefinite block matrices and their blocks.

1. Introduction

We denote by the vector space of all complex matrices. For , the conjugate transpose of is denoted by . The notation is used to mean that is positive semidefinite. If is a Hermitian element of , then we enumerate its eigenvalues as . The singular values of are enumerated as . These are the eigenvalues of the positive semidefinite matrix . Throughout this paper, we assume that is the positive semidefinite block matrix in the formwhere .

The block matrix , where , is positive partial transpose (i.e., PPT) if both and are positive semidefinite.

For , the norm is called the Fan -norm. The norm is called the spectral norm and the norm is called the trace norm. A norm on is called unitarily invariant if for any and any unitary . Clearly, the spectral norm and the trace norm are unitarily invariant. Recall that a unitarily invariant norm may be considered as defined on for all orders by the rule .

Positive semidefinite matrices partitioned into four blocks play important roles in matrix analysis [1–3] and quantum theory [4, 5]. The related inequalities aroused much interest and several applications were given [6–9]. Of these, the one germane to our discussion occurs in the paper of Ulukök [9]. Ulukök in [9] obtained the following results.

Theorem 1. for .

Theorem 2. for , , such that .

Theorem 3. for .

Theorem 4. Let be a nonnegative increasing continuous concave function on . Then,

Theorem 5. Let be a nonnegative increasing continuous convex function on . Then,

One of the questions that arise from Ulukök’s work is the following. Are the conditions in every inequality essential? Furthermore, it is natural to ask whether stronger inequalities of (2)–(6) might be proved. This is the motivation for our study.

In this paper, we present a refinement of inequality (2) and a generalization of inequality (5). Next we derive a result related to inequality (3) and give a new proof of inequality (6). Additionally, we construct some counterexamples to show that the conditions in (2)–(6) are necessary.

2. Main Result

We begin our discussion with inequality (2). We firstly give an example to show inequality (2) is not always true for .

Example 1. Take in inequality (2) and let with , , and . Then,It is known that [6] if and only if and . Using this, we see inequality (2) is not always true for .
Next, we use block matrix technique to derive some inequalities related to positive semidefinite matrices.

Theorem 6. Let be PPT. Then,for .

Proof. The idea of proof is similar to that in [9], Theorem 3.3. It suffices to show . In [7], Hiroshima proved that for PPT matrix and any unitarily invariant norm.
Let be PPT, where are matrices with rows and columns. Using Hiroshima’s result, we obtain .

Example 2. Let and with , , , and in inequality (3). A calculation shows thatHence,Inequality (3) is violated in this case.

Lemma 7 (see [6]). Let . Then,for such that .

Theorem 8. Let be positive semidefinite and let . Then,for such that .

Proof. Let such that . Then, by Lemma 7,This completes the proof.

Remark 9. Inequality (3) is a quick consequence of Theorem 8 by using the convexity of .

Example 3. Let and with , , and . By using MATLAB software to calculate, we haveHence, inequality (4) is violated for these matrices and the trace norm when .
We give some unitarily invariant norm inequalities for positive semidefinite block matrices. To achieve our goal, we need the following lemmas.

Lemma 10 (see [6]). Let . Then, .

Let be an element of and be the vectors obtained by rearranging the coordinates of in decreasing order.

Lemma 11 (see [6]). Let be an increasing convex function and with . Then,

Lemma 12 (see [10]). Let and let be a nonnegative concave function on . Then, for all unitarily invariant norms,

Lemma 13 (see [11]). Let be positive semidefinite and let be an increasing nonnegative continuous convex function on . Then,

Theorem 14. Let be positive semidefinite and let be a nonnegative concave function on . Then,for all unitarily invariant norms.

Proof. We need only to prove the theorem when , since the general case follows by a limit argument due to Lee [12].
Notice that for positive definite matrices, singular values and eigenvalues are the same. Since and using the fact includes that [1], we obtainBy Fan’ s dominance principle [6], we obtainwhere the first inequality follows from inequality (19) and the fact that is nondecreasing, the second inequality is due to Lemma 12, and the third inequality follows from Lemma 10.
This completes the proof.

Corollary 15. Let be a nonnegative increasing continuous concave function on . Then,for all unitarily invariant norms.

Example 4. Let with , , , and . Then,which shows inequality (5) is not always true without the condition that is concave.
Finally, we give a new proof of inequality (6).

Theorem 16. Let be positive semidefinite and let be a nonnegative increasing continuous convex function on . Then,

Proof. An application of the polar decomposition reveals , and we see thatUsing the fact implies that [1], and we getBy the spectral mapping theorem, we have for . Since is an increasing convex function, by Lemma 10, Lemma 11, and Lemma 13 and Fan’ s dominance principle [6], we obtainInequality (27) is equivalent toThis completes the proof.

Example 5. Let with , , , and . As we see, is concave. By computation,Hence,for the trace norm, which shows that inequality (6) is not always true if is not a convex function.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.