Topological classification of sesquilinear forms: reduction to the nonsingular case

Two sesquilinear forms $Phi:mathbb C^mtimesmathbb C^mto mathbb C$ and $Psi:mathbb C^ntimesmathbb C^nto mathbb C$ are called topologically equivalent if there exists a homeomorphism $varphi :mathbb C^mto mathbb C^n$ (i.e., a continuous bijection whose inverse is also a continuous bijection) such that $Phi(x,y)=Psi(varphi (x),varphi (y))$ for all $x,yin mathbb C^m$. R.A.Horn and V.V.Sergeichuk in 2006 constructed a regularizing decomposition of a square complex matrix $A$; that is, a direct sum $SAS^*=Roplus J_{n_1}oplusdotsoplus J_{n_p}$, 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 $Phi$ and $Psi$ 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 $Rinmathbb C^{rtimes r}$ by a nonsingular matrix $Rinmathbb C^{rtimes r}$ such that $R$ and $R$ are the matrices of topologically equivalent forms. Analogous results for real and complex bilinear forms are also obtained.