Signal Processing and Linear Systems-B.P.Lathi copy

79 b73 because a satisfies t he c haracteristic eq b77

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Unformatted text preview: or I f we mUltiply b oth sides by A, t he l eft-hand side is An +l, a nd t he r ight-hand side contains t he t erms An, A n-I, . .. , A. Using Eq. (B.80), i f we s ubstitute An in terms of An -1, An -2, . .. , A, t he h ighest power on t he r ight-hand side is reduced t o n - l. C ontinuing in this way, we see t hat An +k c an b e e xpressed in terms of A n-I, A n-2, . .. , A for a ny k. Hence, t he infinite series on t he r ight-hand s ide of Eq. (B.79) can always b e expressed in t erms of An -1, An -2, . .. , A as (B.75) (B.81) T he s olution f or t his set of homogeneous equations exists if a nd only if IA - >.11 = 1>.1 - AI = 0 (B.80) (B.76a) I f we assume t hat t here a re n d istinct eigenvalues A I, A2, . .. , An, t hen Eq. (B.81) holds for these n values of A. T he s ubstitution o f these values in Eq. (B.81) yields n s imultaneous equations 44 Background • Al At An - 1 [30 A2 1 f (>'l) A~ An - 1 45 B.6 Vectors a nd M atrices E xample B .l3 Let us consider the case where [31 f (A2) 1 2 (B.82a) . ..................... 1 / (An) . .. A2 n An n An - 1 [3n-1 The eigenvalues are a nd [30 [31 Al A2 1 An - 1 - 1 1 /(A1) A2 1 A2 2 An - 1 /(A2) 2 (B.82b) Hence, A1 = - 1, >'2 = - 2, and . ..................... [3n-1 1 An A2 n ... n An - 1 / (An) in which Since A also satisfies Eq. (B.80), we m ay advance a similar argument t o show t hat if / (A) is a f unction o f a s quare m atrix A expressed as a n infinite power series in A , t hen [30] = [ (31 00 f (A) = a ol + a1A + a 2A2 + ... + ... = L :aiAi = t -1 1 (B.83a) i =O [1 -1] [e[2 -1] [e1 a nd -1 e - 2t -2 t e - 2t [2e- et = 2t _ e-t _ e - 2t (B.83b) and in which t he coefficients [3is a re found from Eq. (B.82b). I f some of t he eigenvalues are r epeated ( multiple roots), t he r esults are somewhat modified. We shall d emonstrate t he u tility of t his r esult with the following two examples. B.6-5 2 e - t - e - 2t Computation o f an Exponential and a Power o f a Matrix = [ _ 2e- t L et us c ompute e At defined by (B.84) + 2 e- 2t • Computation o f A k As Eq. (B.83b) indicates, we c an express A k as From Eq. (B.83b), we c an express n -1 e At = A k = [30 1 + [31 A L: [3i(A)i i =l in which t he !3is a re given by Eq. (B.82b), w ith / (Ai) = e A". + ., . + [3n-1 A n -1 in which t he [3iS a re given by Eq. (B.82b) with / (Ai) = A~. F or a completed exampl, of t he c omputation of A k b y t his m ethod, see Example 13.12. 46 Background B.7 Miscellaneous 47 B.7 Miscellaneous B.7-5 Complex Numbers B.1-1 L'Hopital's Rule I f l imf(x)/g(x) r esults in t he i ndeterministic form % or 00/00, t hen n even n odd , f (x) - I' j (x) I1m g{x) - 1m g{x) B.7-2 e ±j9 a +jb=re j9 The Taylor and Maclaurin Series f {x) = f {a) f {x) = f(O) B.7-3 (x - a) , I! { re j9 )k (x - a)2 .. 2! + - - f ( a ) + - - f ( a ) + ... x2 x. = cos IJ ± j s in IJ r =v'a 2 +b 2 , = r k e jk9 ( rl ej9, ) (r2 ej92 ) + -! f{O) + -2! f{O) + ... I B. 7-6 Trigonometric Identities Power Series eX x2 2! x3 3! e ±jx xn n! + - + - +...
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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

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