k and by 736 f a i1 f i gi where gi

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: rs http://www.amazon.com/exec/obidos/ASIN/0898714540 At first glance this definition seems to have an edge over the infinite series approach because there are no convergence issues to deal with. But convergence worries have been traded for uniqueness worries. Because P is not unique, it’s not apparent that (7.3.4) is well defined. The eigenvector matrix P you compute for a given A need not be the same as the eigenvector matrix I compute, so what insures that your f (A) will be the same as mine? The spectral theorem (p. 517) does. Suppose there are k distinct eigenvalues that are grouped according to repetition, and expand (7.3.4) just as (7.2.11) is expanded to produce ⎛ T⎞ Y1 ⎛ f (λ )I 0 ··· 0 ⎞⎜ T ⎟ 1 ⎟ ⎜ f (λ2 )I · · · 0 ⎟⎜ Y2 ⎟ ⎜0 ⎟ ⎜ −1 f (A) = PDP = X1 |X2 | · · · |Xk ⎜ . . . ⎟⎜ .. ⎝. . ⎠⎜ . ⎟ . ⎟ . . . . ⎜.⎟ .⎠ 0 0 · · · f (λk )I ⎝ k D E T H k T f (λi )Xi Yi = = i=1 f (λi )Gi . i=1 T Yk IG R Since Gi is the projector onto N (A − λi I) along R (A − λi I), Gi is uniquely determined by A. Therefore, (7.3.4) uniquely defines f (A) regardless of the choice of P. We can now make a formal definition. Functions of Diagonalizable Matrices Y P Let A = PDP−1 be a diagonalizable matrix where the eigenvalues in D = diag (λ1 I, λ2 I, . . . , λk I) are grouped by repetition. For a function f (z ) that is defined at each λi ∈ σ (A) , define O C f (A) = Pf (D)P−1 ⎛ f (λ )I 1 ⎜0 = P⎜ . ⎝. . 0 0 f (λ2 )I . . . 0 ··· ··· .. . 0 0 . . . ⎞ ⎟ −1 ⎟ P (7.3.5) ⎠ · · · f (λk )I = f (λ1 )G1 + f (λ2 )G2 + · · · + f (λk )Gk , (7.3.6) where Gi is the ith spectral projector as described on pp. 517, 529. The generalization to nondiagonalizable matrices is on p. 603. The discussion of matrix functions was initiated by considering infinite series, so, to complete the circle, a formal statement connecting infinite series with ∞ n (7.3.5) and (7.3.6) is needed. By replacing A by PDP−1...
View Full Document

Ask a homework question - tutors are online