Symmetric functions and the Vandermonde matrix

• Published in: Journal of computational and applied mathematics Vol. 172; no. 1; pp. 49 - 64
• Main Authors: Oruç, Halil, Akmaz, Hakan K.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01-11-2004; Elsevier
• Subjects: Algebra; Algebraic combinatorics; Combinatorics; Combinatorics. Ordered structures; Exact sciences and technology; Linear and multilinear algebra, matrix theory; Mathematics; q-Stirling numbers; Sciences and techniques of general use; Symmetric functions; Triangular and bidiagonal factorization; Vandermonde matrix; q-Stirling numbers; Triangular and bidiagonal factorization; Symmetric functions; Vandermonde matrix; Triangular matrix; Numerical analysis; Symmetric function; Combinatorial identity; Applied mathematics; Wandermonde matrix
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2004.01.032

This work deduces the lower and the upper triangular factors of the inverse of the Vandermonde matrix using symmetric functions and combinatorial identities. The
L and
U matrices are in turn factored as bidiagonal matrices. The elements of the upper triangular matrices in both the Vandermonde matrix and its inverse are obtained recursively. The particular value
x
i
=1+
q+⋯+
q
i−1
in the indeterminates of the Vandermonde matrix is investigated and it leads to
q-binomial and
q-Stirling matrices. It is also shown that
q-Stirling matrices may be obtained from the Pascal matrix.