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A block Hankel generalized confluent Vandermonde matrix. (English) Zbl 1292.15030

Summary: Vandermonde matrices are well known. They have a number of interesting properties and play a role in (Lagrange) interpolation problems, partial fraction expansions, and finding solutions to linear ordinary differential equations, to mention just a few applications. Usually, one takes these matrices square, \(q \times q\) say, in which case the \(i\)-th column is given by \(u(z_i)\), where we write \(u(z) = (1, z, \dots, z^{q - 1})^\top\). If all the \(z_i\) (\(i = 1, \dots, q\)) are different, the Vandermonde matrix is non-singular, otherwise not. The latter case obviously takes place when all \(z_i\) are the same, z say, in which case one could speak of a confluent Vandermonde matrix. Non-singularity is obtained if one considers the matrix \(V(z)\) whose i-th column (\(i = 1, \dots, q\)) is given by the \((i - 1)\)-th derivative \(u^{(i - 1)}(z)^\top\). We consider generalizations of the confluent Vandermonde matrix \(V(z)\) by considering matrices obtained by using as building blocks the matrices \(M(z) = u(z) w(z)\), with \(u(z)\) as above and \(w(z) = (1, z, \dots, z^{r - 1})\), together with its derivatives \(M^{(k)}(z)\). Specifically, we look at matrices whose \(ij\)-th block is given by \(M^{(i + j)}(z)\), where the indices \(i, j\) by convention have initial value zero. These in general non-square matrices exhibit a block-Hankel structure. We answer a number of elementary questions for this matrix. What is the rank? What is the null-space? Can the latter be parametrized in a simple way? Does it depend on \(z\)? What are left or right inverses? It turns out that answers can be obtained by factorizing the matrix into a product of other matrix polynomials having a simple structure. The answers depend on the size of the matrix \(M(z)\) and the number of derivatives \(M^{(k)}(z)\) that is involved. The results are obtained by mostly elementary methods, no specific knowledge of the theory of matrix polynomials is needed.

MSC:

15B05 Toeplitz, Cauchy, and related matrices
15A09 Theory of matrix inversion and generalized inverses
15A23 Factorization of matrices
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References:

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