Normal matrices and polar decompositions in indefinite inner products

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.