An Extension of the Geronimus Transformation for Orthogonal Matrix Polynomials on the Real Line Articles uri icon

publication date

  • December 2016

start page

  • 5009

end page

  • 5032

issue

  • 6

volume

  • 13

international standard serial number (ISSN)

  • 1660-5446

electronic international standard serial number (EISSN)

  • 1660-5454

abstract

  • We consider matrix polynomials orthogonal with respect to a sesquilinear form <., .>(W), such that < P(t)W(t), Q(t)W(t)>(W) = integral(J) P(t)d(mu)Q(t)(T), P, Q is an element of P-pxp [t], where mu is a symmetric, positive definite matrix of measures supported in some infinite subset J of the real line, and W(t) is a matrix polynomial of degree N. We deduce the integral representation of such sesquilinear forms in such a way that a Sobolev-type inner product appears. We obtain a connection formula between the sequences of matrix polynomials orthogonal with respect to mu and <., .>(W), as well as a relation between the corresponding block Jacobi and Hessenberg type matrices.

keywords

  • Matrix orthogonal polynomials; Block Jacobi matrices; Matrix Geronimus transformation; Block Cholesky decomposition; Block LU decomposition; Quasi-determinants; Differential-equations