On some trace inequalities for positive definite Hermitian matrices
© Zhou; licensee Springer. 2014
Received: 14 August 2013
Accepted: 23 January 2014
Published: 12 February 2014
Let A be a positive definite Hermitian matrix, we investigate the trace inequalities of A. By using the equivalence of the deformed matrix, according to some properties of positive definite Hermitian matrices and some elementary inequalities, we extend some previous works on the trace inequalities for positive definite Hermitian matrices, and we obtain some valuable theory.
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries that is equal to its own conjugate transpose. That is, the elements in the i th row and j th column are equal to the complex conjugates of the elements in the j th row and i th column. In other words, the matrix A is Hermitian if and only if , where denotes the conjugate transpose of matrix A.
Hermitian matrices play an important role in statistical mechanics , engineering; in cases such as communication, to describe n-dimensional signal cross-correlation properties, like conjugate symmetry, we can use Hermitian matrices.
Since then, the problems of the trace inequality for positive definite (semidefinite) Hermitian matrices have caught the attention of scholars, getting a lot of interesting results. There exists a vast literature that studies the trace (see [4–8]). In the paper, using the identical deformation of matrix, and combined with some elementary inequalities, our purpose is to derive some new results on the trace inequality for positive definite Hermitian matrices.
The rest of this paper is organized as follows. In Section 2, we will give the relevant definitions and properties of Hermitian matrices. In Section 3, we will quote some lemmas; in Section 4, which is the main part of the paper, using the properties of Hermitian matrices, we investigate the trace inequalities for positive definite Hermitian matrices.
By we denote the n-by-m matrices over a field F, usually the real numbers R or the complex numbers C. Most often, the facts discussed are valid in the setting of the complex-entried matrices, in which case is abbreviated as . In case of square matrices we replace by .
Let ; we may denote the eigenvalues of A by , without loss of generality, where we let . Let denote the singular value, and . It is well known that if A is Hermitian, then all eigenvalues of A are real numbers and if A is unitary, then every eigenvalue of A has modulus 1. The sum of two Hermitian matrices of the same size is Hermitian. If A is Hermitian, then is Hermitian for all . If A is invertible as well, then is Hermitian.
A Hermitian matrix is said to be positive semidefinite, denoted by , if for all , and it is called positive definite, denoted by , if for all nonzero ( denotes complex vector spaces), where denotes the Euclidean inner product on .
The sum of any two positive definite matrices of the same size is positive definite.
Each eigenvalue of a positive definite matrix is a non-negative (positive) real number.
The trace and the determinant of a positive definite matrix are non-negative (positive) real numbers.
Any principal submatrix of a positive definite matrix is positive definite.
A Hermitian matrix is positive definite if and only if all of its eigenvalues are non-negative (positive) real numbers.
We will use this fact several times.
Let . Then the trace of A is given by . The trace function has the following properties.
3 Some lemmas
4 Main results
Now, we prove the following theorem.
Thus, the proof is completed. □
Next, we give a trace inequality for positive definite matrices.
Now we use mathematical induction to deduce our third result.
Then the inequality holds.
That is, the inequality holds when . Thus we have finished the proof. □
I would like to thank the referees for their valuable comments and important suggestions. Project was supported by Hunan Provincial Natural Science Foundation of China #13JJ3118.
- Bebiano N, Da Providencia J, Lemos R: Matrix inequalities in statistical mechanics. Linear Algebra Appl. 2004, 376: 265–273.MATHMathSciNetView ArticleGoogle Scholar
- Minc H, Marcus M: A Survey of Matrix Theory and Matrix Inequalities. Prindle, Weber & Schmidt, Boston; 1964.MATHGoogle Scholar
- Bellman R: Some inequalities for positive definite matrices. In Proceedings of the International Conference on General Inequalities. Birkhäuser, Basel; 1980:89–90.View ArticleGoogle Scholar
- Chan NN, Kwong MK: Hermitian matrix inequalities and a conjecture. Am. Math. Mon. 1985,92(8):533–541. 10.2307/2323157MATHMathSciNetView ArticleGoogle Scholar
- Petz D: survey of certain trace inequalities. In: Functional Analysis and Operator Theory (Warsaw, 1992). Banach Center Publications, vol. 30, pp. 287-298. Polish Acad. Sci., Warsaw (1994)Google Scholar
- Carlen E: Trace inequalities and quantum entropy: an introductory course. In: Entropy and the Quantum, Arizona School of Analysis with Applications, University of Arizona, 16-20 March 2009. Contemporary Mathematics. Am. Math. Soc., Providence (2010)Google Scholar
- Oliveira RI: Sums of random Hermitian matrices and an inequality by Rudelson. Electron. Commun. Probab. 2010, 15: 203–212.MATHMathSciNetView ArticleGoogle Scholar
- Ulukök Z, Türkmen R: On some matrix trace inequalities. J. Inequal. Appl. 2010., 2010: Article ID 201486Google Scholar
- Wang B: Majorization. Beijing Normal University Publishing Group, Beijing; 1990.Google Scholar
- Mitrinovic DS, Vasic PP: Analytic Inequalities. Springer, Berlin; 1970.MATHView ArticleGoogle Scholar
- Magnus JR:A representation theorem for . Linear Algebra Appl. 1987, 95: 127–134.MATHMathSciNetView ArticleGoogle Scholar
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