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On Fischer-type determinantal inequalities for accretive-dissipative matrices
Journal of Inequalities and Applications volume 2015, Article number: 194 (2015)
Abstract
This paper aims to give some refinements of recent results on Fischer-type determinantal inequalities for accretive-dissipative matrices.
1 Introduction
Let \(M_{n} (C)\) be the set of \(n\times n\) complex matrices. For any \(A\in M_{n} (C)\), the conjugate transpose of A is denoted by \(A^{\ast}\). \(A\in M_{n} (C)\) is accretive-dissipative if it has the Hermitian decomposition
where both matrices B and C are positive definite. Conformally partition A, B, C as
such that all diagonal blocks are square. Say k and l (\(k,l>0\) and \(k+l=n\)) the order of \(A_{11}\) and \(A_{22}\), respectively, and let \(m=\min\{ k,l\}\). In this article, we always partition A as in (1.2).
If \(B=I_{n} \) in (1.1), then an accretive-dissipative matrix \(A\in M_{n} (C)\) is called a Buckley matrix.
Let \(A=\bigl({\scriptsize\begin{matrix} {A_{11} } & {A_{12} } \cr {A_{21} } & {A_{22} }\end{matrix}} \bigr)\in M_{n} (C)\). If \(A_{11} \) is invertible, then the Schur complement of \(A_{11} \) in A is denoted by \(A/A_{11} :=A_{22} -A_{21} A_{{11}}^{-1} A_{12} \). For a nonsingular matrix A, its condition number is denoted by \(k(A):=\sqrt{\frac{\lambda_{\mathrm{max}} (A^{\ast}A)}{\lambda _{\mathrm{min}} (A^{\ast}A)}} \), which is the ratio of the largest and the smallest singular value of A. For Hermitian matrices \(B,C\in M_{n} (C)\), we write \(B>(\ge )\, C\) to mean that \(B-C\) is Hermitian positive (semi)definite.
If \(A\in M_{n} (C)\) is positive definite, then the famous Fischer-type determinantal inequality ([1], p.478) states that
If \(A\in M_{n} (C)\) is accretive-dissipative, Ikramov [2] first proved the determinantal inequality
If \(A\in M_{n} (C)\) is accretive-dissipative, Lin [3] proved the determinantal inequality
Recently, Fu and He ([4], Theorem 1) got a stronger result than (1.5) as follows.
Let \(A\in M_{n} (C)\) be accretive-dissipative and partitioned as in (1.2). Then
where \(k=\max(k(B),k(C))\).
For Buckley matrices, Ikramov [2] obtained the stronger bound
In this paper, we will give refinements of (1.6) and (1.7) in Section 2. Other related studies of the Fischer-type determinantal inequalities for accretive-dissipative matrices can be found in [5–7].
2 Main results
We begin this section with the following lemmas.
Lemma 1
([8], Property 6)
Let \(A\in M_{n} (C)\) be accretive-dissipative and partitioned as in (1.2). Then \(A/A_{11}\) is also accretive-dissipative.
Lemma 2
([2], Lemma 1)
Let \(A\in M_{n} (C)\) be accretive-dissipative as in (1.1). Then
Lemma 3
([9], Lemma 3.2)
Let \(B,C\in M_{n} (C)\) be Hermitian and assume B is positive definite. Then
Lemma 4
([10], (6))
Let \(B=\bigl( {\scriptsize\begin{matrix} {B_{11} } & {B_{12} }\cr {B_{12}^{\ast}} & {B_{22} }\end{matrix}} \bigr)\) be Hermitian positive definite. Then
Lemma 5
([3], Lemma 6)
Let \(B,C\in M_{n} (C)\) be positive semidefinite. Then
Lemma 6
Let \(a,b>0\). Then
Lemma 7
Let \(B,C\in M_{n} (C)\) be positive definite. Then
where \(r= \displaystyle{\max_{1\le j\le n}} \textstyle{\{\sqrt{1+\frac{2}{2+(\ln \lambda_{j} )^{2}}} \}}\), \(\lambda_{j} \) are the eigenvalues of \(B^{-1/2}CB^{-1/2}\), and \(B^{1/2}\) means the unique positive definite square root of B.
Proof
Letting \(a=\lambda_{j} \), \(b=\frac{1}{a} \) in Lemma 6 gives \(1+\lambda_{j}\le\sqrt{1+\frac{2}{2+(\ln\lambda_{j})^{2}}} \vert {1+i\lambda_{j} } \vert \), \(j = 1, \ldots,n\). Then
This completes the proof. □
Theorem 1
Let \(A\in M_{n} (C)\) be accretive-dissipative and partitioned as in (1.2). Then
where \(r= \displaystyle{\max_{1\le j\le n}} \textstyle{\{\sqrt{1+\frac{2}{2+(\ln \lambda_{j} )^{2}}} \}}\), \(\lambda_{j} \) are the eigenvalues of \(B^{-1/2}CB^{-1/2}\), \(B^{1/2}\) means the unique positive definite square root of B, and \(k=\max(k(B),k(C))\).
Proof
By Lemma 2 and Lemma 3, we have
with
Set \(A/A_{11} =R+iS\) with \(R=R^{\ast}\) and \(S=S^{\ast}\). By Lemma 1, we obtain
It can be proved that
Thus,
As B, C are positive definite, by Lemma 4, we have
Without loss of generality, we assume \(m=l\), then
where \(k=\max(k(B),k(C))\).
The proof is completed by noting \(\vert {\det A} \vert =\vert {\det A_{11} } \vert \cdot \vert {\det(A/A_{11} )} \vert \). □
Remark 1
Because of \(r\le\sqrt{2}\), inequality (2.1) is a refinement of inequality (1.6).
Theorem 2
Let \(A\in M_{n} (C)\) be accretive-dissipative and partitioned as in (1.2) with \(B_{12} =0\). Then
Proof
Compute
This completes the proof. □
Remark 2
It is clear that inequality (2.5) is an extension of inequality (1.7).
References
Horn, RA, Johnson, CR: Matrix Analysis. Cambridge University Press, London (1985)
Ikramov, KD: Determinantal inequalities for accretive-dissipative matrices. J. Math. Sci. (N.Y.) 121, 2458-2464 (2004)
Lin, M: Fischer type determinantal inequalities for accretive-dissipative matrices. Linear Algebra Appl. 438, 2808-2812 (2013)
Fu, X, He, C: On some Fischer-type determinantal inequalities for accretive-dissipative matrices. J. Inequal. Appl. 2013, 316 (2013)
Lin, M: Reversed determinantal inequalities for accretive-dissipative matrices. Math. Inequal. Appl. 12, 955-958 (2012)
Drury, SW, Lin, M: Reversed Fischer determinantal inequalities. Linear Multilinear Algebra 62, 1069-1075 (2014)
Yang, J: Some determinantal inequalities for accretive-dissipative matrices. J. Inequal. Appl. 2013, 512 (2013)
George, A, Ikramov, KD: On the properties of accretive-dissipative matrices. Math. Notes 77, 767-776 (2005)
Zhan, X: Computing the extremal positive definite solutions of a matrix equation. SIAM J. Sci. Comput. 17, 1167-1174 (1996)
Zhang, F: Equivalence of the Wielandt inequality and the Kantorovich inequality. Linear Multilinear Algebra 48, 275-279 (2001)
Zou, L, Jiang, Y: Improved arithmetic-geometric mean inequality and its application. J. Math. Inequal. 9, 107-111 (2015)
Acknowledgements
The authors wish to express their heartfelt thanks to the referees for their detailed and helpful suggestions for revising the manuscript. At the same time, we are grateful to Prof. Xiaorong Gan for her fruitful discussions.
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Xue, J., Hu, X. On Fischer-type determinantal inequalities for accretive-dissipative matrices. J Inequal Appl 2015, 194 (2015). https://doi.org/10.1186/s13660-015-0721-5
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DOI: https://doi.org/10.1186/s13660-015-0721-5