The sum of two ϕ_S orthogonal matrices when S^−TS is normal and −1 ∉ σ(S^−TS)

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• 作者：Daryl Q. Granario^a ; ^{dgranario@math.upd.edu.ph" class="auth_mail" title="E-mail the corresponding author} ; Dennis I. Merino^b ; ^{dmerino@selu.edu" class="auth_mail" title="E-mail the corresponding author} ; Agnes T. Paras^a ; ^{agnes@math.upd.edu.ph" class="auth_mail" title="E-mail the corresponding author}
• 关键词：15A21 ; 15A23
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：15 April 2016
• 年：2016
• 卷：495
• 期：Complete
• 页码：67-89
• 全文大小：483 K

文摘

Let a nonsingular S∈M_n(C)$S\in M_n\left(\mathbb{C}\right)$ be given. For A∈M_n(C)$A\in M_n\left(\mathbb{C}\right)$, set ϕ_S(A)=S⁻¹A^TS${\varphi }_S\left(A\right)=S^{-1}{A}^{T}S$. We say that A   is a661e" title="Click to view the MathML source">ϕ_S${\varphi }_S$symmetric   if ϕ_S(A)=A${\varphi }_S\left(A\right)=A$; we say that A   is a661e" title="Click to view the MathML source">ϕ_S${\varphi }_S$orthogonal   if A∈GL_n$A\in {\mathit{GL}}_n$ and ϕ_S(A)=A⁻¹${\varphi }_S\left(A\right)=A^{-1}$; we say that A   has a a661e" title="Click to view the MathML source">ϕ_S${\varphi }_S$polar decomposition   if 15ad63466b44cd5b5e" title="Click to view the MathML source">A=UP$A=UP$ for some a661e" title="Click to view the MathML source">ϕ_S${\varphi }_S$ orthogonal U   and a661e" title="Click to view the MathML source">ϕ_S${\varphi }_S$ symmetric P  . Suppose that S^−TS$S^{-T}S$ is normal and −1∉σ(S^−TS)$-1\notin \sigma \left(S^{-T}S\right)$. We determine conditions on A∈M_n(C)$A\in M_n\left(\mathbb{C}\right)$ so that A   can be written as a sum of two a661e" title="Click to view the MathML source">ϕ_S${\varphi }_S$ orthogonal matrices.