Abstract

We introduce the notion of \(J\)-Hermitianity of a matrix, as a generalization of Hermitianity, and, more generally, of closure by \(J\)-Hermitianity of a set of matrices. Many well known algebras, like upper and lower triangular Toeplitz, Circulants and \(\tau \) matrices, as well as certain algebras that have dimension higher than the matrix order, turn out to be closed by \(J\)-Hermitianity. As an application, we generalize some theorems about displacement decompositions presented in [1, 2], by assuming the matrix algebras involved closed by \(J\)-Hermitianity. Even if such hypothesis on the structure is not necessary in the case of algebras generated by one matrix, as it has been proved in [3], our result is relevant because it could yield new low complexity displacement formulas involving not one-matrix-generated commutative algebras.

中文翻译：

---

位移公式中由 J 厄米封闭的代数

摘要

我们引入了矩阵的\(J\) -Hermitianity的概念，作为 Hermitianity 的推广，更一般地说，是一组矩阵的\(J\) -Hermitianity 的闭包。许多著名的代数，如上三角和下三角 Toeplitz、循环和\(\tau \)矩阵，以及维数高于矩阵阶数的某些代数，结果证明是由\(J\) -Hermitianity 封闭的。作为一个应用，我们通过假设所涉及的矩阵代数被\(J\)封闭，从而概括了 [1, 2] 中提出的一些关于位移分解的定理-厄密性。即使在由一个矩阵生成的代数的情况下，这种结构假设不是必需的，正如在 [3] 中所证明的那样，我们的结果是相关的，因为它可以产生新的低复杂度位移公式，而不涉及一个矩阵生成交换代数。