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arXiv · 1507.02959

Using $q$-calculus to study $LDL^T$ 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 coefficients $v_{i,j}=q^{ij}$. We prove that the matrix $L$ is given as a product of diagonal matrices and the lower triangular Toeplitz matrix $T_q$ with coefficients $t_{i,j}=1/(q;q)_{i-j}$, 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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Alexey Kuznetsov. 2015-07-02. Using $q$-calculus to study $LDL^T$ factorization of a certain Vandermonde matrix. https://arxiv.org/abs/1507.02959

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