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Unformatted text preview: for all scalars x, we would like our definitions of sin A and cos A to result in the analogous matrix identity sin2 A + cos2 A = I for all square matrices A. The entrywise approach (7.3.1) clearly fails in this regard. One way to define matrix functions possessing properties consistent with their scalar counterparts is to use infinite series expansions. For example, consider the exponential function ∞ ez = k=0 T H D E zk z2 z3 =1+z+ + ···. k! 2! 3! IG R (7.3.2) Formally replacing the scalar argument z by a square matrix A ( z 0 = 1 is replaced with A0 = I ) results in the infinite series of matrices e A A2 A3 =I+A+ + ···, 2! 3! (7.3.3) Y P called the matrix exponential. While this results in a matrix that has properties analogous to its scalar counterpart, it suffers from the fact that convergence must be dealt with, and then there is the problem of describing the entries in the limit. These issues are handled by deriving a closed form expression for (7.3.3). If A is diagonalizable, then A = PDP−1 = P diag (λ1 , . . . , λn ) P−1 , and k A = PDk P−1 = P diag λk , . . . , λk P−1 , so n 1 O C ∞ eA = k=0 Ak = k! ∞ k=0 PDk P−1 =P k! ∞ k=0 Dk k! P−1 = P diag eλ1 , . . . , eλn P−1 . In other words, we don’t have to use the infinite series (7.3.3) to define eA . Instead, define eD = diag (eλ1 , eλ2 , . . . , eλn ), and set eA = PeD P−1 = P diag (eλ1 , eλ2 , . . . , eλn ) P−1 . This idea can be generalized to any function f (z ) that is defined on the eigenvalues λi of a diagonalizable matrix A = PDP−1 by defining f (D) to be f (D) = diag (f (λ1 ), f (λ2 ), . . . , f (λn )) and by setting f (A) = Pf (D)P−1 = P diag (f (λ1 ), f (λ2 ), . . . , f (λn )) P−1 . Copyright c 2000 SIAM (7.3.4) Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html It is illegal to print, duplicate, or distribute this material Please report violations to meyer@ncsu.edu Buy from AMAZON.com 526 Chapter 7 Eigenvalues and Eigenvecto...
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This document was uploaded on 03/06/2014 for the course MA 5623 at City University of Hong Kong.

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