### Rank of Recurrence Matrices

#### Abstract

A recurrence relation is an equation that recursively defines a sequence of numbers, given one or more initial terms. An m x n recurrence matrix is a matrix whose entries read row-by-row are the terms of a sequence defined by a recurrence relation. The rank of a matrix is the maximum number of linearly independent columns or rows of the matrix. In 2014, Christopher Lee and Valerie Peterson proved that the maximum rank of a recurrence matrix is the order of the corresponding recurrence relation but that for order-two recurrence relations the rank drops if the ratio of the two initial terms of the sequence is an eigenvalue of the relation. Using the method of fundamental solutions, we generalize their result for order-two relations by showing that rank also drops if the n-th powers of the eigenvalues coincide. We then discuss more recent results by Sebastian Bozlee that determine the rank of recurrence matrices based on whether the relation can be written to have lower order and whether the n-th powers of the eigenvalues coincide in hopes of extending our results to recurrence relations of orders higher than two.

#### Keywords

Linear Algebra, Recurrence Matrices, Rank, Recurrence Relations

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