Explicit formulas for the constituent matrices. Application to the matrix functions

Abstract

We present a constructive procedure for establishing explicit formulas of the constituents matrices. Our approach is based on the tools and techniques from the theory of generalized Fibonacci sequences. Some connections with other results are supplied. Furthermore,we manage to provide tractable expressions for the matrix functions, and for illustration purposes we establish compact formulas for both the matrix logarithm and the matrix pth root. Some examples are also provided.

Keywords

• Generalized Fibonacci sequences
• Binet formula
• Constituents matrices
• matrix function
• Matrix logarithm
• Matrix pth root