- Open Access
An extension of the Golden-Thompson theorem
© Li and Zhao; licensee Springer. 2014
- Received: 5 July 2013
- Accepted: 12 December 2013
- Published: 9 January 2014
In this paper, we shall prove for normal matrices A, B. In particular, if A, B are Hermitian matrices, yielding the Golden-Thompson inequality.
MSC:15A16, 47A63, 15A45.
- normal matrix
- Golden-Thompson inequality
The famous Golden-Thompson inequality [1–4] for Hermitian matrices A, B states that . This inequality is a basic tool in quantum statistical mechanics and extensions to infinite dimension have an extensive literature [5, 6]. In this paper, we extend the classical Golden-Thompson theorem to normal matrices.
Throughout this paper, we adopt the following notation. Let be the set of all complex matrices. For a matrix , as usual, its conjugate transpose is denoted by . A matrix A is called Hermitian if , normal if , and unitary if ( is the identity matrix of order n). Given a matrix , the eigenvalues and singular values of A are denoted by , and , respectively, where and . In particular, when A is positive semidefinite (), then . For simplicity, we denote and . Recall that the singular values of a matrix are defined to be the eigenvalues of , i.e., . Here is the spectral norm of A. It is known that the spectral norm over is unitarily invariant, i.e., for all unitary matrices U, V.
The formula above implies that .
In this section, we shall propose some lemmas, laying the foundations of our main results in the next section.
Lemma 2.1 
Here note that is the spectral norm of X.
as desired. □
Thus we get for integers , as desired. □
Here we note that holds for any normal matrix.
The following lemma needs the notion of the Grassmann power (or antisymmetric tensor product), which can be found in [, p.18].
which, equivalently, says that . This completes the proof. □
In this section, we shall present the main results of this paper.
This completes the proof. □
Thus, we can get the following corollary by using the Fan Dominance Principle [, p.56].
for all unitarily invariant norms .
From Theorem 3.1, we can also have the following result.
Proof By Weyl’s majorant theorem we have . Hence Theorem 3.1 implies the desired inequality in Theorem 3.3. □
for Hermitian matrices A, B.
So we get the desired inequality. This completes the proof. □
for Hermitian matrices A, B.
The authors would like to thank the referees for reading this work carefully, providing valuable suggestions and comments, which have significantly improved this article. This work is supported by National Natural Science Foundation of China (Grant No. 61379001).
- Golden S: Lower bounds for the Helmholtz function. Phys. Rev. 1965, 137: B1127-B1128. 10.1103/PhysRev.137.B1127MathSciNetView ArticleMATHGoogle Scholar
- Hiai F: Equality cases in matrix norm inequalities of Golden-Thompson type. Linear Multilinear Algebra 1994, 35: 239-249.MathSciNetView ArticleMATHGoogle Scholar
- Symanzik K: Proofs and refinements of an inequality of Feynman. J. Math. Phys. 1965, 6: 1155-1156. 10.1063/1.1704383View ArticleGoogle Scholar
- Thompson CJ: Inequality with applications in statistical mechanics. J. Math. Phys. 1965, 6: 1812-1813. 10.1063/1.1704727View ArticleMathSciNetGoogle Scholar
- Araki H: Golden-Thompson and Peierls-Bogoliubov inequalities for a general von Neumann algebra. Commun. Math. Phys. 1973, 34: 167-178. 10.1007/BF01645678MathSciNetView ArticleMATHGoogle Scholar
- Ruskai MB: Inequalities for traces on von Neumann algebras. Commun. Math. Phys. 1972, 26: 280-289. 10.1007/BF01645523MathSciNetView ArticleMATHGoogle Scholar
- Ando T: Majorizations, doubly stochastic matrices, and comparison of eigenvalues. Linear Algebra Appl. 1989, 118: 163-248.MathSciNetView ArticleMATHGoogle Scholar
- Bhatia R: Matrix Analysis. Springer, Berlin; 1997.View ArticleMATHGoogle Scholar
- Marshall AW, Olkin I: Inequalities: Theory of Majorization and Its Applications. Academic Press, New York; 1979.MATHGoogle Scholar
- Zhan X Lecture Notes in Mathematics 1790. In Matrix Inequalities. Springer, Berlin; 2002.View ArticleGoogle Scholar
- Furuta T: Norm inequalities equivalent to Lowner-Heinz theorem. Rev. Math. Phys. 1989, 1: 135-137. 10.1142/S0129055X89000079MathSciNetView ArticleMATHGoogle Scholar
- Horn RA, Johnson CR: Matrix Analysis. Cambridge University Press, Cambridge; 1985.View ArticleMATHGoogle Scholar
- Zhang F Universitext. In Matrix Theory: Basic Results and Techniques. Springer, New York; 1999.View ArticleGoogle Scholar
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