Page 84 - Textos de Matemática Vol. 44

74 KH. D. IKRAMOV
and the coninvariant subspaces of matrix (1.1). We describe this relation in Section 3 after the required facts about coninvariant subspaces have been pre- sented in Section 2.
2. Coneigenvalues and coninvariant subspaces For a subspace L ⊂ Cn, define
L={x | x∈L},
where x is the component-wise conjugate of the column vector x.
Definition 2.1. L is a coninvariant subspace of A if (2.1) AL ⊂ L.
In particular, if dimL = 1, then every nonzero vector x ∈ L is called a coneigenvector of A.
The following proposition is an immediate consequence of the above defini- tion.
Proposition 2.2. If L ⊂ Cn is a coninvariant subspace of A, then L is an invariant subspace of the matrix
AL = AA, while L is an invariant subspace of the matrix
AR = AA.
Suppose that n × n matrices A and B are related by the equality
(2.2) B = Q−1AQ.
A transformation of this type is called a consimilarity transformation, while A
and B are said to be consimilar to each other.
Proposition 2.3. If L ⊂ Cn is a coninvariant subspace of A, then Q−1L is a
coninvariant subspace of the matrix B in (2.2). Indeed, inclusion (2.1) implies that
(Q−1AQ)(Q−1L) ⊂ Q−1L = Q−1L.
To a great extent, the theory of coninvariant subspaces is parallel to that of invariant subspaces. There is, however, an important distinction: every matrix A ∈ Mn(C) has an invariant subspace of any given dimension from 1 to n − 1.