The solvability conditions of matrix equations with K-involution

Liang, Mao-Lin ; Dai, Li-Fang

The electronic journal of linear algebra, Tome 22 (2011), pp. 1138-1147.

Résumé

Summary: Let m $\times m$ complex matrix P and n $\times n$ complex matrix Q be k-involutions, i.e., Pk - 1 = P - 1 , Q k - 1 = Q - 1 for some integer k $\geq 2$. An m $\times n$ complex matrix A is (P, Q,$ \beta $)- symmetric if P AQ - $1 = \lambda \beta A$ , or (P, Q,$ \alpha , \beta $)-symmetric if P AQ - $\alpha = \lambda \beta A$ , where $\lambda = e 2\pi $i/k and $\alpha , \beta \in {1, 2, . . . , k}$. In this paper, for given matrices X, Y, E, F with appropriate sizes, the solvability of matrix equations AX = E and Y * A= F under (P, Q,$ \beta $)- and (P, Q,$ \alpha , \beta $)-constraints, respectively, are investigated. Meanwhile, the associated optimal approximation problem is also considered when the above P and Q are unitary.

Détail

Classification : 65F05, 15A24
Keywords: k -involution, (P, Q, $ \beta $)-symmetric matrices, (P, Q, $ \alpha $, beta$)-symmetric matrices$, matrix equations, optimal approximation