H-Normal Matrices
Birkhäuser-Verlag eBooks · 2006
The main ideas developed in this chapter are, first, a classification (up to unitary similarity) of normal matrices in an indefinite inner product space and, second, the nature of canonical forms in this classification. In full generality, this problem area remains unsolved. We will show that it is not less complex than the problem of classification of pairwise v x v commuting matrices (up to simultaneous similarity), where v = min(v+, v-) and v+(v-) is the number of positive (negative) eigenvalues of H. In the research literature the latter problem is said to be “wild”. In contrast, it is well known that for the case when H is positive definite the situation is clear and can be reduced to the problem of simultaneous diagonalization of pairwise commuting selfadjoint matrices.A complete theory is presented here for the special case when the indefinite inner product is defined by \([x,y] = (Hx,y)\), where H is invertible and hermitian with only one negative or positive eigenvalue. This includes the description of canonical form and invariants.In this chapter we use consistently the language of linear transformations, i.e., a linear transformation on Cn is identified with its matrix representation in the standard basis e1, . . . , en. On the other hand, it will also be convenient to work with matrix representations of the same linear transformation with respect to different bases.