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

2 2t e e cos t 2e 2t t e t similarly we define

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Unformatted text preview: 211 = 1, ID311 = 1, I Dul a nd IAI e - t +e- 2t O ne of t he c onditions necessary for a unique solution of Eq. (B.61) is t hat t he n umber of e quations m ust equal the number of unknowns. This implies t hat t he m atrix A m ust b e a s quare matrix. In addition, we observe from t he solution as given in Eq. (B.62) t hat if t he solution is t o exist, IAI "lO.t Therefore, t he inverse exists only for a s quare m atrix a nd only under t he condition t hat t he d eterminant of t he m atrix b e nonzero. A m atrix whose determinant is nonzero is a n onsingular m atrix. T hus, a n inverse exists only for a nonsingular (square) matrix. By definition, we h ave • 41 B.6 Vectors a nd Matrices = - 4. = 8, ID221 = - 1, ID321 = - 5, ID121 = -4 ID231 = - 1 ID331 = 3 ID131 j J e -2t dt A(t)dt= [ J et dt J sin dt J (e- t + 2 e- 2t ) dt We can readily prove t he following identities: Therefore, -4 A -1 = -~ 8 [ -4 • t These two c onditions imply t hat t he n umber of equations is equal t o t he n umber of unknowns a nd t hat all t he e quations a re independent. d dA dB (B.71a) ; U(A+B) = d i + d i d dA (B.71b) ;U(cA) = c di d ; U(AB) dA dB· . = d i B + A di = A B + A B (B.71c) 42 B ackground 43 B.6 Vectors a nd M atrices T he proofs of i dentities (B.71a) a nd (B.71b) are trivial. We c an prove Eq. (B.71c) as follows. Let A b e a n m x n m atrix a nd B a n n x p m atrix. T hen, if a ll - A C =AB (B.76b) =0 from Eq. (B.56), we have n C ik = L . .. j -1 a nd E quation (B.76a) [or (B.76b)] is known as t he c haracteristic e quation o f t he m atrix A a nd c an b e expressed as n C ik = L n a ijbjk j -1 + L a ijbjk (B.72) (B.77) j-1 ------- ------dik or a nn - A a ijbjk e ik E quation ( B.72) along w ith t he m ultiplication rule clearly indicate t hat d ik is t he i kth e lement o f m atrix AB a nd e ik is t he i kth element of m atrix AB. E quation (B.71c) t hen follows. I f we l et B = A -I i n Eq. (B.71c), we o btain Q(A) is called t he c haracteristic p olynomial o f t he m atrix A . T he n zeros of t he c haracteristic polynomial are t he eigenvalues of A and, corresponding t o each eigenvalue, t here is a n eigenvector t hat satisfies Eq. (B.74). T he C ayley-Hamilton t heorem s tates t hat every n X n m atrix A satisfies its own characteristic equation. I n o ther words, Eq. (B.77) is valid if A is r eplaced by A: Q (A) = A n + a n -1 A n -1 + ... + a lA + a oA 0 = 0 (B.78) Functions o f a Matrix B ut since T he C ayley-Hamilton t heorem c an be used t o e valuate functions of a square m atrix A , as shown below. Consider a function f(A) in t he form of a n infinite power series: dId - (AA- )=-1=0 dt dt we have (B.79) (B.73) Because A satisfies t he c haracteristic Eq. (B.77), we c an write 8 .6-4 The Characteristic Equation o f a Matrix: The Cayley-Hamilton Theorem For a n (n X n) s quare m atrix A , any vector x (x t 0) t hat satisfies t he e quation A X=AX (B.74) is a n e igenvector (or c haracteristic v ector), a nd A is t he c orresponding e igenvalue (or c haracteristic v alue) of A. E quation (B.74) can b e expressed as ( A - AI)x = 0...
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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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