The Least Squares {P,Q,k+1}-Reflexive Solution to a Matrix Equation

DONG Chang-zhou, LI Hao-xue

doi:10.13371/j.cnki.chin.q.j.m.2023.02.008

Chinese Quarterly Journal of Mathematics >

2023 , Vol. 38 >Issue 2: 210 - 220

DOI: https://doi.org/10.13371/j.cnki.chin.q.j.m.2023.02.008

The Least Squares {P,Q,k+1}-Reflexive Solution to a Matrix Equation

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School of Mathematics and Science, Hebei GEO University, Shijiazhuang, 050031, China;

DONG Chang-zhou, male, native of Shijiazhuang, Hebei, associate professor of Hebei GEO University, engages in matrix algebra; LI Hao-xue (1996-), female, native of Handan, Hebei, master student of Hebei GEO University, engages in matrix algebra

Received date: 2022-03-25

Online published: 2022-10-10

Supported by

Supported by the Education Department Foundation of Hebei Province ( Grant No.
QN2015218).

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Abstract

Let P ∈C ^m×m and Q∈C ^n×n be Hermitian and {k +1 } -potent matrices,
i.e., P ^k+1 = P = P ^∗, Q^k+1 = Q = Q^∗ , where ( · ) ∗ stands for the conjugate transpose of a
matrix. A matrix X ∈C^m×n is called {P,Q,k +1 } -reflexive (anti-reflexive) if PXQ = X
( PXQ = −X ). In this paper, the least squares solution of the matrix equation AXB = C
subject to {P,Q,k +1 } -reflexive and anti--reflexive constraints are studied by converting
into two simpler cases: k=1 and k=2.

Key words： Matrix equations; Potent matrix; {P,Q,k +1 } -reflexive (anti-reflexive);
Canonical correlation decomposition; Least squares solution

Cite this article

DONG Chang-zhou, LI Hao-xue . The Least Squares {P,Q,k+1}-Reflexive Solution to a Matrix Equation[J]. Chinese Quarterly Journal of Mathematics, 2023 , 38(2) : 210 -220 . DOI: 10.13371/j.cnki.chin.q.j.m.2023.02.008

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