TABLE OF CONTENTS Although there are some notable differences between the algebra of scalars and the algebra of matrices, there are many similarities. For example, it has been observed that if A is a square matrix, it may be raised to a power such as
and this adds to considerations of adding exponents, as in
Moreover, the multiplication of A by itself, no matter how many times, is certainly commutative, and if only A is involved, multiplication is distributive, as in
It would seem, therefore, that an n th order matrix polynomial could be defined in the same manner as an m th order algebraic polynomial
This is done in this chapter, as is a study of an infinite series of matrices and matrix functions.
The Cayley-Hamilton Theorem states that a matrix satisfies its own characteristic equation. The proof provides some introductory insight regarding the formulation and manipulation of matrix polynomials.
The characteristic matrix of A has been defined in eq (7.4)
and this matrix possesses an adjoint, B = adjK, which is the transpose of the cofactor matrix, K c. Because A is n n, the elements of K c and of adj K possess a maximum degree of n ? 1. The foregoing remarks concerning the adjoint and the cofactor matrix as well as the proof of the Cayley-Hamilton Theorem can be well developed by means...
