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2018•Operators and MatricesOpen access

Using q-calculus to study LDLt factorization of a certain Vandermonde matrix

Alexey Kuznetsov

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Abstract

We use tools from q-calculus to study LDL T decomposition of the Vandermonde matrix V q with entries v i, j = q i j . We prove that the matrix L is given as a product of diagonal matrices and the lower triangular Toeplitz matrix T q with elements t i, j = 1/(q;q) i-j , where (z;q) k is the q-Pochhammer symbol. We investigate some properties of the matrix T q , in particular, we compute explicitly the inverse of this matrix.

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We use tools from q-calculus to study LDL T decomposition of the Vandermonde matrix V q with entries v i, j = q i j . We prove that the matrix L is given as a product of diagonal matrices and the lower triangular Toeplitz matrix T q with elements t i, j = 1/(q;q) i-j , where (z;q) k is the q-Pochhammer symbol. We investigate some properties of the matrix T q , in particular, we compute explicitly the inverse of this matrix.

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Available abstract

We use tools from q-calculus to study LDL T decomposition of the Vandermonde matrix V q with entries v i, j = q i j . We prove that the matrix L is given as a product of diagonal matrices and the lower triangular Toeplitz matrix T q with elements t i, j = 1/(q;q) i-j , where (z;q) k is the q-Pochhammer symbol. We investigate some properties of the matrix T q , in particular, we compute explicitly the inverse of this matrix.

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

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