半张量积下矩阵方程组AX=B,XC=D的最小二乘解

LEAST-SQUARES SOLUTIONS OF MATRIX EQUATIONS AX=B,XC=D WITH SEMI-TENSOR PRODUCT

本文研究了半张量积下矩阵方程组AX=B,XC=D在不同情况下的最小二乘解X*∈R^(p×q),其中矩阵A∈R^(m×n),B∈R^(h×k),C∈R^(a×b),D∈R^(l×d)给定.根据半张量积的定义将其转变为普通乘积下的矩阵方程组,再结合矩阵奇异值分解及矩阵微分给出该方程组在不同情况下最小二乘解的解析表达式,并用数值算例加以验证.

In this paper, we consider the least-square solutions X＊∈ R^（p×q） of the matrix equations AX = B, XC = D with respect to the semi-tensor product, where matrices A ∈ R^（m×n）, B ∈ R^（h×k）, C ∈ R^（a×b）, D ∈ R^（l×d）are given. By using the definition of the semi-tensor product, we transform the original problem under semi-tensor product into some related matrix least squares with the conventional matrix product. Then, combining with the differentiation and singular value decomposition of matrices, we give the explicit representation of the least squares solution. Finally, we present some elementary numerical examples to verify the proposed results.