Topological classification of sesquilinear forms: Reduction to the nonsingular case

Abstract

Two sesquilinear forms Φ:Cm×Cm→C and Ψ:Cn×Cn→C are called topologically equivalent if there exists a homeomorphism φ:Cm→Cn (i.e., a continuous bijection whose inverse is also a continuous bijection) such that Φ(x,y)=Ψ(φ(x),φ(y)) for all x,y∈Cm. R.A. Horn and V.V. Sergeichuk in 2006 constructed a regularizing decomposition of a square complex matrix A; that is, a direct sum SAS⁎=R⊕Jn1⊕⋯⊕Jnp, in which S and R are nonsingular and each Jni is the ni-by-ni singular Jordan block. In this paper, we prove that Φ and Ψ are topologically equivalent if and only if the regularizing decompositions of their matrices coincide up to permutation of the singular summands Jni and replacement of R∈Cr×r by a nonsingular matrix R′∈Cr×r such that R and R′ are the matrices of topologically equivalent forms Cr×Cr→C. Analogous results for bilinear forms over C and over R are also obtained.