Zhuo-Heng He

Bio: Zhuo-Heng He is an academic researcher from Shanghai University. The author has contributed to research in topics: Quaternion & Quaternion algebra. The author has an hindex of 16, co-authored 36 publications receiving 772 citations. Previous affiliations of Zhuo-Heng He include Katholieke Universiteit Leuven & Auburn University.

A real quaternion matrix equation with applications

Abstract: Let i, j, k be the quaternion units and let A be a square real quaternion matrix. A is said to be η-Hermitian if −η A*η = A, where η ∈ {i, j, k} and A* is the conjugate transpose of A. Denote A η* = − η A*η. Following Horn and Zhang's recent research on η-Hermitian matrices (A generalization of the complex AutonneTakagi factorization to quaternion matrices, Linear Multilinear Algebra, DOI:10.1080/03081087.2011.618838), we consider a real quaternion matrix equation involving η-Hermicity, i.e. where Y and Z are required to be η-Hermitian. We provide some necessary and sufficient conditions for the existence of a solution (X, Y, Z) to the equation and present a general solution when the equation is solvable. We also study the minimal ranks of Y and Z satisfying the above equation.

Some matrix equations with applications

Abstract: We establish necessary and sufficient conditions for the solvability to the matrix equation and present an expression of the general solution to (1) when it is solvable. As applications, we discuss the consistence of the matrix equation where * means conjugate transpose, and provide an explicit expression of the general solution to (2). We also study the extremal ranks of X 3 and X 4 and extremal inertias of and in (1). In addition, we obtain necessary and sufficient conditions for the classical matrix equation to have Re-nonnegative definite, Re-nonpositive definite, Re-positive definite and Re-negative definite solutions. The findings of this article extend related known results. †Dedicated to Professor Ky Fan (1914–2010).

Matrices: Methods and Applications

Abstract: (1992). Matrices: Methods and Applications. Journal of the Operational Research Society: Vol. 43, No. 12, pp. 1185-1185.

Coupled Sylvester-type Matrix Equations and Block Diagonalization

Abstract: We prove Roth-type theorems for systems of matrix equations including an arbitrary mix of Sylvester and $\star$-Sylvester equations, in which the transpose or conjugate transpose of the unknown matrices also appear. In full generality, we derive consistency conditions by proving that such a system has a solution if and only if the associated set of $2 \times 2$ block matrix representations of the equations are block diagonalizable by (linked) equivalence transformations. Various applications leading to several particular cases have already been investigated in the literature, some recently and some long ago. Solvability of these cases follow immediately from our general consistency theory. We also show how to apply our main result to systems of Stein-type matrix equations.