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Some properties of the \(q\)-adic Vandermonde matrix. (English) Zbl 0851.15017
The construction of \(q\)-adic Vandermonde matrices \(W\) by forming a Taylor expansion of a foundation polynomial is described. The change of the basis matrix from the standard ones to the \(q\)-adic coordinates is given. The relation between the \(q\)-adic coordinate columns is studied by introducing the concept of a polynomial in a hypercompanion matrix.
The determinant of \(W\) is calculated. The relation between \(W\) and the hypercompanion matrix is discussed. It is shown that the determinant of \(W\) is a product of resultants. The matrix \(W\) has a block structure that is similar to that of a Wronskian. This block structure is studied using a modified definition of the partial derivative operator.
Reviewer: V.Burjan (Praha)
15B57 Hermitian, skew-Hermitian, and related matrices
15A15 Determinants, permanents, traces, other special matrix functions
15A21 Canonical forms, reductions, classification
41A10 Approximation by polynomials
12Y05 Computational aspects of field theory and polynomials (MSC2010)
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