Functions of Matrices

Abstract

Power series in matrices. Let $P\left( \lambda \right) = \sum\limits_{i = 0}^\infty {{a_i}{\lambda ^i}}$ be an ordinary power series with complex coefficients in the complex variable λ. If for a matrix A of order n with complex elements every element of

$$P_m (A) = \sum\limits_{i = 0}^m {a_i \lambda ^{_i} } $$

approaches a finite limit as m → ∞

$$P(A) = \sum\limits_{i = 0}^\infty {a_i \lambda ^{_i} } $$

the matrix is said to exist and to be equal to the matrix of these limiting values.