Classification of linear mappings between indefinite inner product spaces

Abstract

Let $\mathcal{A}:U\to V$ be a linear mapping between vector spaces $U$ and $V$ over a field or skew field $\mathbb{F}$ with symmetric, or skew-symmetric, or Hermitian forms $\mathcal{B}:U\times U\to \mathbb{F}$ and $\mathcal{C}:V\times V\to \mathbb{F}.$ We classify the triples $(\mathcal{A},\mathcal{B},\mathcal{C})$ if $\mathbb{F}$ is $\mathbb{R}$, or $\mathbb{C}$, or the skew field of quaternions $\mathbb{H}$. We also classify the triples $(\mathcal{A},\mathcal{B},\mathcal{C})$ up to classification of symmetric forms and Hermitian forms if the characteristic of $\mathbb{F}$ is not 2.

21 pages

References in corpus (6)

• Canonical forms for complex matrix congruence and *congruence
• Classification problems for system of forms and linear mappings
• Congruences of a square matrix and its transpose
• Canonical matrices of bilinear and sesquilinear forms
• Computation of the canonical form for the matrices of chains and cycles of linear mappings
• Canonical matrices of isometric operators on indefinite inner product spaces

Cited by in corpus (3)

• Isometric and selfadjoint operators on a vector space with nondegenerate diagonalizable form
• Classification of linear operators satisfying $(Au,v)=(u,A^{r}v)$ or $(Au,A^{r}v)=(u,v)$ on a vector space with indefinite scalar product
• Symplectic spaces and pairs of symmetric and nonsingular skew-symmetric matrices under congruence