一类特殊的Toeplitz矩阵行列式的计算

Calculation of Determinant of a Special Toeplitz Matrix

摘要: Toeplitz矩阵是结构矩阵的一种特殊形式，其研究在矩阵与计算数学理论中占有重要地位，本文主要讨论一类特殊的Toeplitz矩阵行列式的求解，运用行列式的性质得到三个递推关系式，从而将这类特殊的Toeplitz矩阵行列式的求解转化为三对角Toeplitz矩阵行列式的求解，构造等差和类等比数列结合递推关系式求出通项表达式，进而给出了这类特殊的Toeplitz矩阵行列式的精确解。作为应用，解决了这类特殊的Toeplitz矩阵正定性判定的问题。

Abstract: Toeplitz matrix is one of special forms of structural matrix, and its study plays an important role in the theory of matrix and computational mathematics. This paper mainly discusses the solution of the determinant of a special Toeplitz matrix, three recursive relations are obtained by using the properties of the determinant, and the solution of the determinant of the special Toeplitz matrix is transformed into the solution of the determinant of the tridiagonal Toeplitz matrix, the arithmetic difference and quasi-arithmetic sequence are constructed by combining the recursive relations to find the general term expression, and the exact solution of the determinant of the special Toeplitz matrix is given. As application, the positive qualitative determination problem of this special Toeplitz matrix is solved.