Yoopyo Hong proved in 1989 that each Hermitian matrix A is orthogonally *congruent to a matrix of the form ${\epsilon }_1{A}_1\oplus \cdots {\epsilon }_r{A}_r\oplus B_1\oplus \cdots \oplus B_s$, in which $A_1,\dots ,A_r,B_1,\dots ,B_s$ are uniquely determined by the orthogonal *congruence class of A, and ${\epsilon }_1,\dots ,{\epsilon }_r\in \{1,-1\}$. We prove that ${\epsilon }_1,\dots ,{\epsilon }_p$ are uniquely determined by the orthogonal *congruence class of A as well. As an application, we present a canonical form of a pair $(A,B)$ consisting of a Hermitian matrix A and a nonsingular symmetric matrix B with respect to transformations $(S^{⁎}AS,S^{T}BS)$ with a nonsingular S.

中文翻译：

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关于正交*同余的厄米矩阵的 Hong 规范形式

Yoopyo Hong 在 1989 年证明了每个 Hermitian 矩阵A都是正交*全等的矩阵形式${\epsilon }_1{\mathrm{一种}}_1\oplus \cdots {\epsilon }_r{\mathrm{一种}}_r\oplus {乙}_1\oplus \cdots \oplus {乙}_{秒}$，其中 ${\mathrm{一种}}_1,\dots ,{\mathrm{一种}}_r,{乙}_1,\dots ,{乙}_{秒}$ 由 A 的正交 *同余类唯一确定，并且 ${\epsilon }_1,\dots ,{\epsilon }_r\in \{1,-1\}$. 我们证明${\epsilon }_1,\dots ,{\epsilon }_{磷}$也由 A 的正交 *同余类唯一确定。作为一个应用程序，我们提出了一对的规范形式$(\mathrm{一种},乙)$由一个 Hermitian 矩阵A和一个关于变换的非奇异对称矩阵B 组成$({秒}^{⁎}\mathrm{一种}秒,{秒}^{吨}乙秒)$与非奇异S。