Page 85 - Textos de Matemática Vol. 44

SOLUTIONS TO SESQUILINEAR MATRIX EQUATIONS 75
In general, this is not true of coninvariant subspaces. For instance, the skew- symmetric matrix
0 1 (2.3) A= −1 0
has two one-dimensional eigenspaces corresponding to its distinct eigenvalues i and −i. However, A has no one-dimensional coninvariant subspace because the relation
would imply that
Ax = λx ALx = |λ|2x,
which is impossible since
(2.4) AL =AA=A2 =−I2.
To find out what dimension a coninvariant subspace of a matrix A ∈ Mn(C) may have, we first recall the notion of a coneigenvalue. The coneigenvalues of A are the n scalars attributed to A and preserved by any consimilarity transformation of A. Their definition is based on two remarkable properties of the spectrum of AL (or, which is the same, the spectrum of AR):
1. It is symmetric with respect to the real axis. Moreover, the eigenvalues λ and λ¯ are of the same multiplicity, and their partial multiplicities (that is, the orders of the corresponding Jordan blocks) are identical.
2. The negative eigenvalues of AL (if any) are necessarily of even algebraic multiplicity. Moreover, all of their partial multiplicities are even.
For the proofs of these properties, we refer the reader to [1, pp. 252–253]. Let
σ(AL) = {λ1,...,λn}
be the spectrum of AL. The coneigenvalues of A are the n scalars μ1,...,μn defined as follows:
If λi ∈ σ(AL) does not lie on the negative real axis, then the corresponding coneigenvalue μi is defined as the square root of λi with a nonnegative real part and the multiplicity of μi is set equal to that of λi :
μi = λ1/2, Reμi ≥ 0. i
With a real negative λi ∈ σ(AL), we associate two conjugate purely imagi- nary coneigenvalues
μi = ±λ1/2, i
the multiplicity of each being half the multiplicity of λi. The set {μ1, . . . , μn} is called the conspectrum of A.