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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