Explicit formulas for the determinants of Toeplitz matrices

Abstract

We give formulas for the determinant of a Toeplitz matrix for three cases. The first case is for 3 bands. These results use expressions for the power sums and symmetric functions of cosine sequences. The second case is for any symmetric Toeplitz matrix such as a covariance. In this case the determinant has 2 factors. The third case is for any nxn Toeplitz matrix with odd elements zero. In this case its determinant, say d_n, is a perfect square for n even, and d_2n+1 = √d_2nd_2n+2, so again each determinant has 2 factors. We also give some results for the inverse of a Toeplitz matrix. AMS 2000 Subject Classification: 15A15; 15A09.