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Topological classification of sesquilinear forms: reduction to the nonsingular case

Two sesquilinear forms $Φ:\mathbb C^m\times\mathbb C^m\to \mathbb C$ and $Ψ:\mathbb C^n\times\mathbb C^n\to \mathbb C$ are called topologically equivalent if there exists a homeomorphism $φ:\mathbb C^m\to \mathbb C^n$ (i.e., a continuous bijection whose inverse is also a continuous bijection) such that $Φ(x,y)=Ψ(φ(x),φ(y))$ for all $x,y\in \mathbb C^m$. R.A.Horn and V.V.Sergeichuk in 2006 constructed a regularizing decomposition of a square complex matrix $A$; that is, a direct sum $SAS^*=R\oplus J_{n_1}\oplus\dots\oplus J_{n_p}$, in which $S$ and $R$ are nonsingular and each $J_{n_i}$ is the $n_i$-by-$n_i$ 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 $J_{n_i}$ and replacement of $R\in\mathbb C^{r\times r}$ by a nonsingular matrix $R'\in\mathbb C^{r\times r}$ such that $R$ and $R'$ are the matrices of topologically equivalent forms. Analogous results for real and complex bilinear forms are also obtained.