科研之友
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摘要
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 x q say, in which case the i-th column is given by u(z(i)), where we write u(z) = (1, z,..., z(q-1))(T). If all the zi (i = 1,, q) are different, the Vandermonde matrix is non-singular, otherwise not. The latter case obviously takes place when all z 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,..., q) is given by the (i - 1)-th derivative u((i-1)) (z)(T). We will 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,...,z(T-1)), together with its derivatives M-(k) (z). Specifically, we will 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 will 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.
- 出版日期2014-8-15
