# Functions of Matrices

• C. C. Mac Duffee
Chapter
Part of the Ergebnisse der Mathematik und Ihrer Grenƶgebiete book series (MATHE1, volume 5)

## 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.

## Notes

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