矩阵广义khatri-Rao积的一些性质及不等式

Some Properties and Inequalities on the Generalized Khatri-Rao Product of Matrices

在Kronecker积,Hadamared积与Khatri-Rao积的基础上给出了矩阵A,B的广义Khatri-Rao积f(A,B)的定义.并给出了广义Khatri-Rao积f(A,B)的一些普遍性质,得到正定矩阵、半正定矩阵、非负矩阵、Hermite矩阵的广义Khatri-Rao积的特殊性质.推出了广义Khatri-Rao积的共轭转置矩阵运算结果.证明了逆矩阵、平方矩阵的广义Khatri-Rao积的几个重要不等式以及半正定矩阵的广义Khatri-Rao积特征值的性质.

Based on the definition of Kronecker product, Hadamard product and Khatri-Rao product, the generalized Khatri-Rao product of matrices A and B, written asf（A ,B）, are defined. Some common properties on the generalized Khatri-Rao product and some special properties on generalized Khatri-Rao product of the specific matrices, such as positive matrix, semi-positive matrix, nonnegative matrix, Hermite matrix are presented. The operation of conjugate transpose on the generalized Khatri-Rao product is deduced out. Moreover, some important inequalities about the generalized Khatri-Rao product of the inverse matrices and square matrices are proved. At last, properties on the eigenvalue of the generalized Khatri-Rao product of the semi-matrices are presented.