We show that using Dunford-Taylor’s integral, a classical tool of functional analysis, it is possible to derive an expression for the inverse of a general non-singular complex-valued tridiagonal matrix. The special cases of Jacobi’s symmetric and Toeplitz (in particular symmetric Toeplitz) matrices are included. The proposed method does not require the knowledge of the matrix eigenvalues and relies only on the relevant invariants which are determined, in a computationally effective way, by means of a dedicated recursive procedure. The considered technique has been validated through several test cases with the aid of the computer algebra program Mathematica©.

中文翻译：

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使用Dunford-Taylor积分对三对角矩阵求逆

我们表明，使用经典的泛函分析工具Dunford-Taylor积分，可以为一般的非奇异复值三对角矩阵的逆导出一个表达式。包括Jacobi对称矩阵和Toeplitz（特别是对称Toeplitz）矩阵的特殊情况。所提出的方法不需要矩阵特征值的知识并且仅依赖于相关的不变量，这些不变量以计算有效的方式借助于专用的递归过程来确定。在计算机代数程序Mathematica©的帮助下，已通过多个测试案例对所考虑的技术进行了验证。