Topological classification of sesquilinear forms: reduction to the nonsingular case

Two sesquilinear forms $\mathrm{\Phi }:{\mathbb{C}}^m\times {\mathbb{C}}^m\to \mathbb{C}$ and $\mathrm{\Psi }:{\mathbb{C}}^n\times {\mathbb{C}}^n\to \mathbb{C}$ are called topologically equivalent if there exists a homeomorphism $\varphi :\mathbb C^m\to \mathbb C^n$ (i.e., a continuous bijection whose inverse is also a continuous bijection) such that $\mathrm{\Phi }(x,y)=\mathrm{\Psi }(\phi (x),\phi (y))$ 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 $SA{S}^{\ast }=R\oplus J_{n_1}\oplus \cdots \oplus J_{n_p}$, 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 $\mathrm{\Phi }$ and $\mathrm{\Psi }$ 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 $R\in\mathbb C^{r\times r}$byanonsingularmatrix$R’\in\mathbb C^{r\times r}$suchthat$R$and$R’$ are the matrices of topologically equivalent forms. Analogous results for real and complex bilinear forms are also obtained.