Minghua Lin

An eigenvalue majorization inequality for positive semidefinite block matrices

Let be a Hermitian matrix. It is known that the vector of diagonal elements of H, diag(H), is majorized by the vector of the eigenvalues of H, λ(H), and that this majorization can be extended to the eigenvalues of diagonal blocks of H. Reverse majorization results for the eigenvalues are our goal. Under the additional assumptions that H is positive semidefinite and the block K is Hermitian, the main result of this article provides a reverse majorization inequality for the eigenvalues. This results in the following majorization inequalities when combined with known majorization inequalites on the left:

A matrix trace inequality and its application

In this short paper, we give a complete and affirmative answer to a conjecture on matrix trace inequalities for the sum of positive semidefinite matrices. We also apply the obtained inequality to derive a kind of generalized Golden-Thompson inequality for positive semidefinite matrices.

A pretty binomial identity

Elementary proofs abound: the first identity results from choosing x = y = 1 in the binomial expansion of (x+y). The second one may be obtained by comparing the coefficient of x in the identity (1 + x)(1 + x) = (1 + x). The reader is surely aware of many other proofs, including some combinatorial in nature. At the end of the previous century, the evaluation of these sums was trivialized by the work of H. Wilf and D. Zeilberger [8]. In the preface to the charming book [8], the authors begin with the phrase You’ve been up all night working on your new theory, you found the answer, and it is in the form that involves factorials, binomial coefficients, and so on, ... and then proceed to introduce the method of creative telescoping discussed in Section 3. This technique provides an automatic tool for the verification of these type of identities. The points of view presented in [3] and [10] provide an entertaining comparison of what is admissible as a proof. In this short note we present a variety of proofs of the identity

On a determinantal inequality arising from diffusion tensor imaging

In comparing geodesics induced by different metrics, Audenaert formulated the following determinantal inequality

Regions of convergence of a Padé family of iterations for the matrix sector function and the matrix pth root

\det(A^2+|BA|)\le \det(A^2+AB),

A property of the geometric mean of accretive operators

where

Inverses and eigenvalues of diamond alternating sign matrices

A, B

Reversed Fischer determinantal inequalities

are