Topological classification of sesquilinear forms: Reduction to the nonsingular case

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• 作者：Carlos M. da Fonseca^a ; ^{carlos@sci.kuniv.edu.kw" class="auth_mail" title="E-mail the corresponding author} ; Tetiana Rybalkina^b ; ^{rybalkina_t@ukr.net" class="auth_mail" title="E-mail the corresponding author} ; Vladimir V. Sergeichuk^b ; ^{sergeich@imath.kiev.ua" class="auth_mail" title="E-mail the corresponding author}
• 关键词：15A21 ; 15B33 ; 37J40
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：1 September 2016
• 年：2016
• 卷：504
• 期：Complete
• 页码：581-589
• 全文大小：297 K

文摘

Two sesquilinear forms Φ:C^m×C^m→C$\mathrm{\Phi }:{\mathbb{C}}^m\times {\mathbb{C}}^m\to \mathbb{C}$ and Ψ:Cⁿ×Cⁿ→C$\mathrm{\Psi }:{\mathbb{C}}^n\times {\mathbb{C}}^n\to \mathbb{C}$ are called topologically equivalent if there exists a homeomorphism 52c44d" title="Click to view the MathML source">φ:C^m→Cⁿ$\phi :{\mathbb{C}}^m\to {\mathbb{C}}^n$ (i.e., a continuous bijection whose inverse is also a continuous bijection) such that Φ(x,y)=Ψ(φ(x),φ(y))$\mathrm{\Phi }(x,y)=\mathrm{\Psi }(\phi (x),\phi (y))$ for all x,y∈C^m$x,y\in {\mathbb{C}}^m$. 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⊕J_n1⊕⋯⊕J_np$SA{S}^{⁎}=R\oplus J_{n_1}\oplus \cdots \oplus J_{n_p}$, in which S and R   are nonsingular and each J_ni$J_{n_i}$ is the n_i$n_i$-by-n_i$n_i$ 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_ni$J_{n_i}$ and replacement of R∈C^r×r$R\in {\mathbb{C}}^{r\times r}$ by a nonsingular matrix c40a84ca3db8e4fd1ad94126d2a9c1d" title="Click to view the MathML source">R^′∈C^r×r$R^{\prime }\in {\mathbb{C}}^{r\times r}$ such that R   and R^′$R^{\prime }$ are the matrices of topologically equivalent forms C^r×C^r→C${\mathbb{C}}^r\times {\mathbb{C}}^r\to \mathbb{C}$. Analogous results for bilinear forms over C$\mathbb{C}$ and over R$\mathbb{R}$ are also obtained.