ABSTRACT

In this work, we explore an extension of the Omega calculus in the context of matrix analysis introduced recently by Neto [Matrix analysis and Omega calculus. SIAM Rev. 2020;62(1):264–280]. We obtain Omega representations of analytic functions of three important classes of matrices: companion, tridiagonal, and triangular. Our representation recovers the main results of Chen and Louck [The combinatorial power of the companion matrix. Linear Algebra Appl. 1996;232:261–278] on the powers of the companion matrix. Furthermore, we generalize previous work on the powers of tridiagonal matrices due to Gutiérrez–Gutiérrez in [Powers of tridiagonal matrices with constant diagonals. Appl Math Comput. 2008;206(2):885–891], Öteleş and Akbulak [Positive integer powers of certain complex tridiagonal matrices. Appl Math Comput. 2013;219(21):10448–10455], and triangular matrices following Shur [A simple closed form for triangular matrix powers. Electron J Linear Algebra. 2011;22:1000–1003].

摘要

在这项工作中，我们探索了 Omega 微积分在 Neto 最近引入的矩阵分析背景下的扩展 [矩阵分析和 Omega 微积分。暹罗修订版 2020；62(1):264–280]。我们获得了三类重要矩阵的解析函数的 Omega 表示：伴矩阵、三对角矩阵和三角矩阵。我们的表示法恢复了 Chen 和 Louck 的主要结果 [伴随矩阵的组合能力。线性代数应用。1996;232:261–278] 关于伴随矩阵的权力。此外，由于 Gutiérrez–Gutiérrez 在 [具有恒定对角线的三对角矩阵的幂。应用数学计算。2008;206(2):885–891], Öteleş 和 Akbulak [某些复数三对角矩阵的正整数幂。应用数学计算。2013;219(21):10448–10455], 和三角矩阵遵循 Shur [三角矩阵幂的简单闭合形式。电子 J 线性代数。2011；22：1000–1003]。