The nth Power of 2x2 Matrices Applications to Second-Order Linear Recurrences and Birth–Death Processes
Arbai Aziz *
ENSIT - Ecole des Nouvelles Sciences d’ing´enierie, Le Laboratoire Syst´emes, Contrˆole et D´ecision (LSCD), Tanger, Morocco.
*Author to whom correspondence should be addressed.
Abstract
After exploring a robust method for calculating the nth power of a 2x2 matrix An without resorting to diagonalization techniques, we have provided a unique formula for the nth power of any type of 2x2 matrix, diagonalizable or not, with or without two distinct eigenvalues, and whether these eigenvalues are real or not. In this article, we will discuss applications of the above, such as:
1) Application of the nth power of 2x2 matrices to solve second-order linear recurrence problems;
2) The (unsolved) problem of the birth and death process (BDP) with constant coefficients and an infinite number of states. That is to say we will apply our matrix power formula to determine the eigenvectors of the generating matrix A of this process.
Keywords: Power of a matrix, recurrent linear sequences of order 2, the birth and death process