Page 83 - Textos de Matemática Vol. 44

SOLUTIONS TO SESQUILINEAR MATRIX EQUATIONS: CONSPECTRAL APPROACH
KHAKIM D. IKRAMOV
Dedicated to Nat´alia Bebiano on the occasion of her 60th birthday.
Abstract. There is a well-known relation between the solutions to the qua- dratic matrix equation
XDX + AX + XB + C = 0
with n×n matrix coefficients and the n-dimensional invariant subspaces of the
2n × 2n matrix
We establish a similar relation between the solutions to the sesquilinear matrix
−B −D M=CA.
equation
and the n-dimensional coninvariant subspaces of M.
1. Introduction
Let Mn(C) be the set of n×n complex matrices. Set m = 2n, and let A, B, C and D be given matrices in Mn(C). There is a well-known relation between the solutions to the quadratic matrix equation
XDX + AX + XB + C = 0
and the n-dimensional invariant subspaces of the m × m matrix
−B −D (1.1) M= C A.
Our goal in this communication is to establish a similar relation between the solutions to the sesquilinear matrix equation
(1.2) XDX + AX + XB + C = 0
2010 Mathematics Subject Classification. 15A21
Key words and phrases. sesquilinear matrix equation; consimilarity transformation; coneigen- value; coninvariant subspace; partial multiplicity
73
XDX + AX + XB + C = 0