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Unformatted text preview: or I f we mUltiply b oth sides by A, t he l efthand side is An +l, a nd t he r ighthand side
contains t he t erms An, A nI, . .. , 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 ighthand 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 nI, A n2,
. .. , A for a ny k. Hence, t he infinite series on t he r ighthand 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 [3n1 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 . ..................... [3n1 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 _ et _ 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.65 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". + ., . + [3n1 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.75 Complex Numbers B.11 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.72 e ±j9 a +jb=re j9 The Taylor and Maclaurin Series f {x) = f {a)
f {x) = f(O) B.73 (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. 76 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.
 Spring '13
 Bayliss
 Signal Processing, The Land

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