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# Some determinantal inequalities for accretive-dissipative matrices

*Journal of Inequalities and Applications*
**volume 2013**, Article number: 512 (2013)

## Abstract

This article aims to obtain some determinantal inequalities for accretive-dissipative matrices which are generalizations of the determinantal inequalities presented by Lin (Linear Algebra Appl. 438:2808-2812, 2013). At the same time, we give some numerical examples which show the effectiveness of our results.

## 1 Introduction

Let {\mathbb{M}}_{n}(\mathbf{C}) be the space of complex matrices of size n\times n matrices. A\in {\mathbb{M}}_{n}(\mathbf{C}) is said to be accretive-dissipative, if, in its Toeplitz decomposition

both matrices *B* and *C* are Hermitian positive definite. For simplicity, let *A*, *B*, *C* be partitioned as

such that the diagonal blocks {A}_{11} and {A}_{22} are of order *k* and *l* (k>0, l>0 and k+l=n), respectively, and let m=min\{k,l\}.

If A\in {\mathbb{M}}_{n}(\mathbf{C}) is partitioned as

where {A}_{11} is a nonsingular submatrix, then the matrix A/{A}_{11}:={A}_{22}-{A}_{21}{A}_{11}^{-1}{A}_{12} is called the *Schur complement* of the submatrix {A}_{11} in *A*.

If A\in {\mathbb{M}}_{n}(\mathbf{C}) is positive definite and partitioned as in (1.2), then the inequalities [[1], Lemma 6] hold:

If A\in {\mathbb{M}}_{n}(\mathbf{C}) is positive definite and partitioned as in (1.2), then the famous Fischer-type determinantal inequality is proved [[2], p.478]:

If A\in {\mathbb{M}}_{n}(\mathbf{C}) is an accretive-dissipative matrix and partitioned as in (1.2), Ikramov [3] first proved the determinantal inequality for *A*:

Lin [[1], Theorem 8] got a stronger result than (1.5) as follows:

If A\in {\mathbb{M}}_{n}(\mathbf{C}) is an accretive-dissipative matrix, then

The purpose of this paper is to give some generalizations of (1.3) and (1.6). Our main results can be stated as follows.

**Theorem 1** *Let* B,C\in {\mathbb{M}}_{n}(\mathbf{C}) *be positive definite and* *x*, *y* *be positive real numbers*. *Then*

When x=y, the inequality det(B+C)\le {2}^{\frac{n}{2}}|det(B+iC)| is a special case of Theorem 1. Thus (1.5) is a generalization of the inequality |det(B+iC)|\le det(B+C)\le {2}^{\frac{n}{2}}|det(B+iC)| [[1], Lemma 6].

**Theorem 2** *Let* A\in {\mathbb{M}}_{n}(\mathbf{C}) *be accretive*-*dissipative and partitioned as in* (1.2), *and let* *x*, *y* *be positive real numbers*. *Then*

When x=y, we get the inequality |detA|\le {2}^{\frac{n}{2}}|det{A}_{11}||det{A}_{22}| [[1], (3.1)], which is a special case of Theorem 2.

**Theorem 3** *Let* A\in {\mathbb{M}}_{n}(\mathbf{C}) *be accretive*-*dissipative and partitioned as in* (1.2), *and let* *x*, *y* *be positive real numbers*. *Then*

When x=y, we get the inequality [[1], (3.2)]

which is a special case of (1.7).

## 2 Proofs of main results

To achieve the proofs of Theorem 1, Theorem 2 and Theorem 3, we need the following lemmas.

**Lemma 4** [[4], Property 6]

*Let* A\in {\mathbb{M}}_{n}(\mathbf{C}) *be accretive*-*dissipative and partitioned as in *(1.2). *Then* A/{A}_{11}:={A}_{22}-{A}_{21}{A}_{11}^{-1}{A}_{12}, *the Schur complement of* {A}_{11} *in* *A*, *is also accretive*-*dissipative*.

**Lemma 5** [[3], Lemma 1]

*Let* A\in {\mathbb{M}}_{n}(\mathbf{C}) *be accretive*-*dissipative and partitioned as in* (1.2). *Then* {A}^{-1}=E-iF *with* E={(B+C{B}^{-1}C)}^{-1} *and* F={(C+B{C}^{-1}B)}^{-1}.

**Lemma 6** [[3], Lemma 4]

*Let* B,C\in {\mathbb{M}}_{n}(\mathbf{C}) *be Hermitian and assume that* B>0. *Then*

**Remark 1** A stronger inequality than (2.4) was given in Lin [[5], Lemma 2.2]: Let A>0 and any Hermitian *B*. Then A\mathrm{\u266f}(B{A}^{-1}B)\ge B.

*Proof of Theorem 1* Let {\lambda}_{j}, j=1,\dots ,n, be the eigenvalues of {B}^{-\frac{1}{2}}C{B}^{-\frac{1}{2}}, where {B}^{\frac{1}{2}} means the unique positive definite square root of *B*. Then we have

The first inequality follows from [[6], Theorem 2.2], while the second one we prove is as follows:

The proof is completed. □

*Proof of Theorem 2*

The proof is completed. □

*Proof of Theorem 3* By Lemma 4, we obtain

Furthermore, by Lemma 5, we have

where {E}_{k} and {F}_{k} are positive definite.

By a simple computation, we obtain

By Lemma 4, it is easy to know that *R*, *S* are positive definite and we have

By the inequalities

it can be proved that

Thus

By Lemma 6 and the operator reverse monotonicity of the inverse, we get

As *B*, *C* are positive definite, we also have

Without loss of generality, assume m=l. Then we have

By noting

the proof is completed. □

## 3 Numerical examples

There are many upper bounds for the determinant of the accretive-dissipative matrices which are due to (1.6), (1.8) and (1.9). However, these bounds are incomparable.

In this section, we give some numerical examples to show that (1.8) and (1.9) are better than (1.6) in some cases.

**Example 3.1** Let

We calculate that |detA|=4.0402.

By the upper bound of |detA| in (1.6), we have

where {A}_{11}=1.01+1.01i, {A}_{22}=1.01+1.01i.

Let x=4, y=5. From the upper bound of |detA| in (1.8), we have

where {B}_{11}=1.01, {B}_{22}=1.01, {C}_{11}=1.01, {C}_{22}=1.01.

Meanwhile, by the upper bound of |detA| in (1.9), we get

**Example 3.2** Let

We calculate that |detA|=119.0378.

By the upper bound of |detA| in (1.6), we have

where {A}_{11}=\left(\begin{array}{cc}5+5i& -1+3i\\ -1+3i& 2+2i\end{array}\right), {A}_{22}=3+4i.

Let x=4, y=5. Then, by the upper bound of |detA| in (1.8), we have

where {B}_{11}=\left(\begin{array}{cc}5& -1\\ -1& 2\end{array}\right), {C}_{11}=\left(\begin{array}{cc}5& 3\\ 3& 2\end{array}\right), {B}_{22}=3, {C}_{22}=4.

Meanwhile, by the upper bound of |detA| in (1.9), we get

From the two examples above, we can obtain that (1.8) and (1.9) are better than (1.6) in some cases.

## References

Lin M: Fisher type determinantal inequalities for accretive-dissipative matrices.

*Linear Algebra Appl.*2013, 438: 2808–2812. 10.1016/j.laa.2012.11.016Horn RA, Johnson CR:

*Matrix Analysis*. Cambridge University Press, London; 1985.Ikramov KD: Determinantal inequalities for accretive-dissipative matrices.

*J. Math. Sci. (N.Y.)*2004, 121: 2458–2464.George A, Ikramov KD: On the properties of accretive-dissipative matrices.

*Math. Notes*2005, 77: 767–776. 10.1007/s11006-005-0077-0Lin M: Notes on an Anderson-Taylor type inequality.

*Electron. J. Linear Algebra*2013, 26: 63–70.Zhan X: Singular values of differences of positive semidefinite matrices.

*SIAM J. Matrix Anal. Appl.*2000, 22: 819–823.

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Yang, J. Some determinantal inequalities for accretive-dissipative matrices.
*J Inequal Appl* **2013**, 512 (2013). https://doi.org/10.1186/1029-242X-2013-512

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DOI: https://doi.org/10.1186/1029-242X-2013-512