Author(s) :
Brian Lins,
Patrick Meade,
Christian Mehl
,
Leiba Rodman
The paper is published :
Linear and Multilinear Algebra, 49: 45-89, 2001
MSC 2000
- 15A63 Quadratic and bilinear forms, inner products
-
15A23 Factorization of matrices
Abstract :
Normal matrices with respect to indefinite inner products are studied using the additive decomposition into selfadjoint
and skewadjoint parts. In particular, several structural properties of indecomposable normal matrices are obtained. These
properties are used to describe classes of matrices that are logarithms of selfadjoint or normal matrices. In turn, we use logarithms
of normal matrices to study polar decompositions with respect to indefinite inner products. It is proved, in particular, that every
normal matrix with respect to an indefinite inner product defined by an invertible Hermitian matrix having at most two negative (or
at most two positive) eigenvalues, admits a polar decomposition. Previously known descriptions of indecomposable normals in
indefinite inner products with at most two negative eigenvalues play a key role in the proof. Both real and complex cases are
considered.
Keywords :
Indefinite inner product, normal matrix, polar decomposition