On the Determinants and Inverses of Circulant Matrices with a General Number Sequence
Provides theoretical results for a class of structured matrices, but the contribution is incremental as it generalizes existing work on circulant matrices with special number sequences.
This paper derives formulas for the determinants and inverses of circulant matrices whose entries follow a generalized second-order linear recurrence. The results extend known formulas for specific sequences like Fibonacci numbers.
The generalized sequence of numbers is defined by W_{n}=pW_{n-1}+qW_{n-2} with initial conditions W_{0}=a and W_{1}=b for a,b,p,q\inZ and n\geq2, respectively. Let W_{n}=circ(W_{1},W_{2},...,W_{n}). The aim of this paper is to establish some useful formulas for the determinants and inverses of W_{n} using the nice properties of the number sequences. Matrix decompositions are derived for W_{n} in order to obtain the results.