- Research Article
- Open Access
On Some Matrix Trace Inequalities
© Z. Ulukök and R. Türkmen. 2010
- Received: 23 December 2009
- Accepted: 14 March 2010
- Published: 6 April 2010
We first present an inequality for the Frobenius norm of the Hadamard product of two any square matrices and positive semidefinite matrices. Then, we obtain a trace inequality for products of two positive semidefinite block matrices by using block matrices.
- Positive Real Number
- Complex Matrice
- Real Vector
- Complex Matrix
- Positive Definite Matrix
Denote and eigenvalues and singular values of matrix , respectively. Since is Hermitian matrix, its eigenvalues are arranged in decreasing order, that is, and if is any matrix, its singular values are arranged in decreasing order, that is, The trace of a square matrix (the sum of its main diagonal entries, or, equivalently, the sum of its eigenvalues) is denoted by .
A norm on is called unitarily invariant for all and all unitary .
As is well known, yields (see, e.g., [1, pages 17–19]).
where is a positive integer.
for and positive definite matrices, where is any positive integer.
Lemma 2.1 (see, e.g., ).
For any and .
Lemma 2.2 (see, e.g., ).
Lemma 2.3 (Cauchy-Schwarz inequality).
Lemma 2.4 (see, e.g., [8, page 269]).
Lemma 2.5 (see, e.g., [9, page 177]).
Lemma 2.6 (see, e.g., ).
where is a positive integer.
for all and all unitarily invariant norms .
By the following theorem, we present an inequality for Frobenius norm of the power of Hadamard product of two matrices.
Thus, by using Theorem 3.1, the desired is obtained.
Now, we give a trace inequality for positive semidefinite block matrices.
where is an integer.
This study was supported by the Coordinatorship of Selçuk University's Scientific Research Projects (BAP).
- Zhan X: Matrix Inequalities, Lecture Notes in Mathematics. Volume 1790. Springer, Berlin, Germany; 2002:viii+116.Google Scholar
- Zhang F: Matrix Theory: Basic Results and Techniques, Universitext. Springer, New York, NY, USA; 1999:xiv+277.View ArticleGoogle Scholar
- Yang X: A matrix trace inequality. Journal of Mathematical Analysis and Applications 2000, 250(1):372–374. 10.1006/jmaa.2000.7068MATHMathSciNetView ArticleGoogle Scholar
- Yang XM, Yang XQ, Teo KL: A matrix trace inequality. Journal of Mathematical Analysis and Applications 2001, 263(1):327–331. 10.1006/jmaa.2001.7613MATHMathSciNetView ArticleGoogle Scholar
- F. M. Dannan, Matrix and operator inequalities., Journal of Inequalities in Pure and AppliedMathematics, vol. 2, no. 3, article 34, 7 pages, 2001.Google Scholar
- Zhang FZ: Another proof of a singular value inequality concerning Hadamard products of matrices. Linear and Multilinear Algebra 1988, 22(4):307–311. 10.1080/03081088808817843MATHMathSciNetView ArticleGoogle Scholar
- Z. P. Yang and X. X. Feng, A note on the trace inequality for products of Hermitian matrix power, Journal of Inequalities in Pure and Applied Mathematics, vol. 3, no. 5, article 78, 12 pages, 2002.Google Scholar
- Lieb EH, Thirring W: Studies in Mathematical Physics, Essays in Honor of Valentine Bartmann. Princeton University Press, Princeton, NJ, USA; 1976.Google Scholar
- Horn RA, Johnson CR: Topics in Matrix Analysis. Cambridge University Press, Cambridge, UK; 1991:viii+607.MATHView ArticleGoogle Scholar
- Wang BY, Gong MP: Some eigenvalue inequalities for positive semidefinite matrix power products. Linear Algebra and Its Applications 1993, 184: 249–260.MATHMathSciNetView ArticleGoogle Scholar
- Horn RA, Mathias R: An analog of the Cauchy-Schwarz inequality for Hadamard products and unitarily invariant norms. SIAM Journal on Matrix Analysis and Applications 1990, 11(4):481–498. 10.1137/0611034MATHMathSciNetView ArticleGoogle Scholar
- Horn RA, Mathias R: Cauchy-Schwarz inequalities associated with positive semidefinite matrices. Linear Algebra and Its Applications 1990, 142: 63–82. 10.1016/0024-3795(90)90256-CMATHMathSciNetView ArticleGoogle Scholar
- Zhang F: Schur complements and matrix inequalities in the Löwner ordering. Linear Algebra and Its Applications 2000, 321(1–3):399–410.MATHMathSciNetView ArticleGoogle Scholar
- Li C-K, Mathias R: Inequalities on singular values of block triangular matrices. SIAM Journal on Matrix Analysis and Applications 2002, 24(1):126–131. 10.1137/S0895479801398517MATHMathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.