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) j1  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 ayleyHamilton 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 ayleyHamilton 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 .64 The Characteristic Equation o f a Matrix: The CayleyHamilton
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.
 Spring '13
 Bayliss
 Signal Processing, The Land

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