Page 89 - Textos de Matemática Vol. 44

SOLUTIONS TO SESQUILINEAR MATRIX EQUATIONS 79
Conversely, assume that L is an M-coninvariant subspace of dimension n and the columns of the matrix
U1 (3.1) UL= U2
form a basis in L. In addition, assume that the n × n block U1 in (3.1) is nonsingular. Note that this assumption is independent of the choice of a basis in L: if VL represents another basis in L, then VL = ULQ for some nonsingular n×n matrix Q. If VL is written as in (3.1), then the block V1 = U1Q is nonsingular.
Since U1 is nonsingular, the matrix
(3.2) X = U U−1
is well defined. From the fact that (3.1) is a base matrix of an A-coninvariant subspace, we deduce that
BU1 + DU2 = −U1ML, CU1 + AU2 = U2ML
for some n × n matrix ML. Multiplying both these relations on the right by U−1, we have
21
1
(3.3) (3.4)
B+DX=−U M U−1, 1L1
C+AX=U M U−1. 2L1
Now, we multiply (3.3) on the left by X and add the resulting equality to (3.4). This yields
XDX + AX + XB + C = 0;
that is, matrix (3.2) is a solution to the sesquilinear equation (1.2). According to Corollary 2.10, matrix (1.1) always has an n-dimensional con- invariant subspace if n is even. Whether or not such a subspace gives rise to a solution to equation (1.2) depends on the nonsingularity of the corresponding
block U1 (see (3.1)).
References
[1] R. A. Horn, C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985.
[2] Y. P. Hong, R. A. Horn, A canonical form of matrices under consimilarity, Linear Algebra Appl. 102 (1988), 143–168.
(Kh. D. Ikramov) Faculty of Computational Mathematics and Cybernetics, Moscow State University, 119992 Moscow, Russia
E-mail address: ikramov@cs.msu.su