Using $q$-calculus to study $LDL^T$ factorization of a certain Vandermonde matrix

TL;DR

This paper employs q-calculus to analyze the LDL^T factorization of a Vandermonde matrix, revealing explicit forms for the factors and properties of the associated Toeplitz matrix.

Contribution

It introduces a novel q-calculus approach to explicitly determine the LDL^T decomposition of a specific Vandermonde matrix and analyzes properties of the resulting Toeplitz matrix.

Findings

Explicit formula for the L matrix as a product involving T_q

Explicit inverse of the T_q matrix

Characterization of T_q's properties

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.