Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6416055 | Linear Algebra and its Applications | 2016 | 9 Pages |
Two sesquilinear forms Φ:CmÃCmâC and Ψ:CnÃCnâC are called topologically equivalent if there exists a homeomorphism Ï:CmâCn (i.e., a continuous bijection whose inverse is also a continuous bijection) such that Φ(x,y)=Ψ(Ï(x),Ï(y)) for all x,yâCm. R.A. Horn and V.V. Sergeichuk in 2006 constructed a regularizing decomposition of a square complex matrix A; that is, a direct sum SASâ=RâJn1ââ¯âJnp, 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 RâCrÃr by a nonsingular matrix Râ²âCrÃr such that R and Râ² are the matrices of topologically equivalent forms CrÃCrâC. Analogous results for bilinear forms over C and over R are also obtained.