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2015•Unpublished venueOpen access

A note on LU decomposition of the Discrete Fourier Transform matrix

Alexey Kuznetsov

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Abstract

We describe some properties of the lower triangular Toeplitz matrix Tq with coefficients ti,j = 1/(q; q)i−j, where (z; q)k is the q-Pochhammer symbol. We identify explicitly the inverse of Tq and show that both this matrix and its transpose appear in LU decomposition of the Vandermonde matrix Vq having coefficients vi,j = q ij. When q is the n-th root of unity, our result gives an explicit LU decomposition of the Discrete Fourier Transform matrix.

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What this paper is about

We describe some properties of the lower triangular Toeplitz matrix Tq with coefficients ti,j = 1/(q; q)i−j, where (z; q)k is the q-Pochhammer symbol. We identify explicitly the inverse of Tq and show that both this matrix and its transpose appear in LU decomposition of the Vandermonde matrix Vq having coefficients vi,j = q ij. When q is the n-th root of unity, our result gives an explicit LU decomposition of the Discrete Fourier Transform matrix.

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Available abstract

We describe some properties of the lower triangular Toeplitz matrix Tq with coefficients ti,j = 1/(q; q)i−j, where (z; q)k is the q-Pochhammer symbol. We identify explicitly the inverse of Tq and show that both this matrix and its transpose appear in LU decomposition of the Vandermonde matrix Vq having coefficients vi,j = q ij. When q is the n-th root of unity, our result gives an explicit LU decomposition of the Discrete Fourier Transform matrix.

Key concepts: Mathematics, Toeplitz matrix, Vandermonde matrix, Matrix (chemical analysis), Inverse, Matrix decomposition, Triangular matrix, Transpose

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