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matrix_exponentials - Notes by David Groisser for MAS 4105...

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Unformatted text preview: Notes by David Groisser for MAS 4105, Copyright c 1998 Exponentials of Matrices. Suppose p is a polynomial of degree m : p ( x ) = b + b 1 x + ... + b m x m for some (real or complex) numbers b ,b 1 ,...,b m . For any n × n square matrix A , we define the n × n matrix p ( A ) by “plugging in A for x ”: p ( A ) = b I + b 1 A + ... + b m A m where I is the n × n identity matrix. To shorten the notation we define A = I and write p ( A ) = ∑ m k =0 b k A k . Example If p ( x ) = 1 + x 2 , and J =- 1 1 . then J 2 =- I , so p ( J ) = 0 0 0 0 . A Taylor series can be thought of as an “infinite polynomial” f ( x ) = ∑ ∞ k =0 b k x k . For any such series, and any n × n matrix A , we can define f ( A ) = ∞ k =0 b k A k provided the series converges (i.e. if for each i and j , the series for the ( ij ) th entry of the series for f ( A ) converges). Note that in general f ( A ) is not the matrix whose ( ij ) th entry is f ( A ij ), so you must be careful with notation: ( f ( A...
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