TABLE OF CONTENTS The development of functions of nondiagonalizable matrices parallels the development for functions of diagonal matrices that was presented in 7.3 except that the Jordan form is used in place of the diagonal matrix of eigenvalues. Recall from the discussion surrounding (7.3.5) on p. 526 that if A ? C n n is diagonalizable, say A = PDP ?1, where D = diag ( ? 1 I, ? 2 I,..., ? s I), and if f ( ? i) exists for each ? i, then f ( A) is defined to be
The Jordan decomposition A = PJP ?1 described on p. 590 easily provides a generalization of this idea to nondiagonalizable matrices. If J is the Jordan form for A, it's natural to define f ( A) by writing f ( A) = P f ( J) P ?1. However, there are a couple of wrinkles that need to be ironed out before this notion actually makes sense. First, we have to specify what we mean by f ( J)-this is not as clear as f ( D) is for diagonal matrices. And after this is taken care of we need to make sure that P f ( J) P ?1 is a uniquely defined matrix. This also is not clear because, as...
