Page 157 - Schaum's Outline of Differential Equations
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CHAPTER 16
At
e
DEFINITION
Fora square matrix A.
The infinite series (16.1) converges for every A and ;. so that e is defined for all square matrices.
COMPUTATION OF e*'
kt
For actual!} computing the elements of e , (16.1) is not generally useful. However, it follows (with some
effort) from Theorem I .1. I. applied to the matrix A/, thai the infinite scries can be reduced to a polynomial in;. Thus:
Theorem 16.1. If A is a malrK ha\ ing n rows and n columns, then
o
where a$. a, «„_, are functions ff which must be determined for each A.
Example 16.1. When A has two rows and two columns, then /) = 2 and
When A has throe rows and three oiilnmiis, ihen n = ^ and
Theorem 16.2, Let A he as in t hcorcm 16.1, and define
Then if ?., is an eigenvalue of A;.
Furthermore, if X ; is an eigeinalue of multiplicity k. k> 1. then the following equations are
also valid:
141)
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