Xingzhi Zhan

Computing the Extremal Positive Definite Solutions of a Matrix Equation

An efficient and numerically stable implementation of a known algorithm is suggested for finding the extremal positive definite solutions of the matrix equation

On the matrix equation X + ATX−1A = I

X+A^*X^{-1}A=I

Inequalities for Unitarily Invariant Norms

, if such solutions exist. The convergence rate is analyzed. A new algorithm that avoids matrix inversion is presented. Numerical examples are given to illustrate the effectiveness of the algorithms.

Extremal Eigenvalues of Real Symmetric Matrices with Entries in an Interval

Simple necessary and sufficient conditions for the matrix equation X + ATX−1A = I to have a positive definite solution are given. These conditions are used to explain some properties of the equation and to obtain new information. Finally we prove a conjecture on the extremal solutions.

Singular Values of Differences of Positive Semidefinite Matrices

Let A,B,X be complex matrices with A,B positive semidefinite. It is proved that (2+t)||ArXB2-r+A2-rXBr||\le 2||A2X+tAXB+XB2|| for any unitarily invariant norm

Inequalities involving unitarily invariant norms and operator monotone functions

||\cdot||

Compact operators whose real and imaginary parts are positive

and real numbers r,t satisfying

Lower bounds for the spread of a Hermitian matrix

1\le 2r\le 3,

Some research problems on the hadamard product and singular values of matrices

-2 < t\le 2.