544 Chapter 19The matrix eigenvalue problem
EXAMPLE 19.10The eigenvectors of the matrix (Examples 19.2 and 19.3)
are (ignoring the arbitrary multipliers)
corresponding to eigenvaluesλ
11 = 1 − 1 , λ
21 = 11 , λ
31 = 12. Then
and
so that AX 1 = 1 XD
0 Exercises 23, 24
If the matrix Xof the eigenvectors of Ais nonsingular then premultiplication of both
sides of equation (19.26) by the inverse matrixX
− 1gives
D 1 = 1 X
− 1AX (19.28)
and Ahas been reduced to the diagonal formD.
XD=−
−
011
123
111
100
010
002
=
−
012
126
112
AX=
−
−
−
−
211
11 4 5
110
011
123
111
=
−
012
126
112
Xxxx==−D
( ) , =
1231011
123
111
λ 0 00
00
00
100
010
002
23λ
λ
=
−
xxx
1230
1
1
1
2
1
1
=−
,=
,= 33
1
A=
−
−
−
211
11 4 5
110