19.4 Matrix diagonalization 543
wherec
2andc
3are arbitrary. The normalized vectors
are also orthogonal; they form an orthonormal set of four-dimensional vectors.
0 Exercises 20, 21
0 Exercise 22
19.4 Matrix diagonalization
Let the square matrix Ahave eigenvectorsx
1, x
2, x
3, =, x
ncorresponding to eigenvalues
λ
1, λ
2, λ
3, =, λ
n:
Ax
k1 = 1 λ
kx
k, k 1 = 1 1, 2, 3, =, n (19.24)
and let Xbe the matrix whose columns are the eigenvectors of A,
(19.25)
Then
AX 1 = 1 (Ax
lAx
2Ax
3- Ax
n)
= 1 (λ
1x
1λ
2x
2λ
3A
3- λ
nx
n) (19.26)
= 1 XD
where Dis the diagonal matrix whose diagonal elements are the eigenvalues of A,
(19.27)
D=λ
λ
λ
λ
12300 0
000
00 0
000
nXxxx x==()
12311 12 13 121 22 23 231nnnxxx x
xxx x
xxxx x
xxx x
nn
nnnn32 3331
23
CC
121
2
1
1
1
1
1
2
1
1
1
1
=
,=
−
−
,=
−
−
,=
−
CC
341
2
1
1
1
1
1
2
1
1
1
−−
1