Linear recurrence relations in the algebra of matrices and applications

Periodic matrix difference equations and companion matrices in blocks: some applications

“…In this subsection, we are interested in the use of the matrix Kronecker product for studying some linear recursive relation in the algebra of square matrices in blocks, and their use for the computation of the powers of matrices in blocks through the generalized Cayley-Hamilton Theorem. In fact, using the product of Kronecker, we extend the results of [3], to the algebra of square matrices in blocks.…”

A new approach for second‐order linear matrix descriptor differential equations of Apostol–Kolodner type

“…The non-zero row of the matrix R is the q -row of matrix L. According to the Proposition 3, it is interesting that the solution of the second order linear matrix differential descriptor system (2) is given by (15) which can be easily solved as a linear non-homogenous first order matrix differential system, see [3][4][5][6][7][8] and [9]. Thus, the solution of (2) is summarized in the next Theorem.…”

“…Among the most simple and fundamental equations are the first‐order linear matrix equations with time‐invariant coefficient, that is, X ′ ( t ) = AX ( t ), with X (0) = X 0 . Apparently, the above system has been extensively studied, and the solution depends closely on the computation of e At ; see some basic references with a particular interest for us and . Moreover, the application of Kronecker matrix products in transforming linear‐matrix equations into corresponding matrix‐vector equations is a well‐established technique, and it has been effectively used for the solution of the first‐order linear matrix equation, that is, FX ′ ( t ) = AX ( t ) + X ( t ) B , for F = I , see and for F ≠ I , det F = 0 see .…”

Exact treatment of operator difference equations with nonconstant and noncommutative coefficients

“…The classical field of difference equations deals with linear or nonlinear equations where the unknown is a real or complex-valued function defined on a countable domain which in turn may be thought of as coincident with N * without loss of generality. These equations are conveniently classified as scalar difference equations [2, chap 2,3,4]- [6] to discriminate them from the so-called matrix difference equations [7], [8] where the unknowns constitute a sequence of d × d matrices with entries in C as well as possibly all the given coefficients.…”