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In recent years, a number of fast algorithms for computing the determinant ofa Toeplitz matrix were developed. The fastest algorithm we know so far is oforder $k^2\log{n}+k^3$, where $n$ is the number of rows of the Toeplitz matrixand $k$ is the bandwidth size. This is possible because such a determinant canbe expressed as the determinant of certain parts of $n$-th power of a related$k \times k$ companion matrix. In this paper, we give a new elementary proof ofthis fact, and provide various examples. We give symbolic formulas for thedeterminants of Toeplitz matrices in terms of the eigenvalues of thecorresponding companion matrices when $k$ is small.

A fast elementary algorithm for computing the determinant of toeplitz matrices