Mathematical Methods for Physicists: A concise introduction - Site Map

MATRIX ALGEBRA
Since
0 1
1
B
@ 1
0
C
A
and
0 1
0
B C
@ 0 A
1
are linearly independent, they are the eigenvectors corresponding to ˆ 5.
For ˆ 1, we have
0
10
1 0 1 0
1 0 1
2 2 0 x 1 0
2x 1 ‡ 2x 2 ‡ 0x 3 0
B
CB
@ 2 2 0 A@
x
C B C B
2 A ˆ @ 0 A or 2x 1 2x 2 ‡ 0x
C B C
@
3 A ˆ @ 0 A:
0 0 4 x 3 0
0x 1 ‡ 0x 2 4x 3 0
Solving this system yields
x 1 ˆ t; x 2 ˆ t; x 3 ˆ 0;
where t is arbitrary. Thus the eigenvectors corresponding to ˆ 1 are non-zero
vectors of the form
0 1 0 1
t 1
B C B C
~X ˆ @ t A ˆ t@
1 A:
0 0
It is easy to check that the three eigenvectors
0 1 0 1 0 1
1
0
1
B C B C B C
~X 1 ˆ @ 1 A; ~X 2 ˆ @ 0 A; ~X 3 ˆ @ 1 A;
0
1
0
are linearly independent. We now form the matrix ~S that has ~X 1 , ~X 2 ,and ~X 3 as its
column vectors:
0 1
1 0 1
B C
~S ˆ @ 1 0 1A:
0 1 0
The matrix S ~ 1 A ~ S ~ is diagonal:
0
10
10
1 0 1
1=2 1=2 0 3 2 0 1 0 1 5 0 0
~S 1 A ~ S ~ B ˆ @ 0 0 1CB
CB
C B C
A@
2 3 0A@
1 0 1A ˆ @ 0 5 0A:
1=2 1=2 0 0 0 5 0 1 0 0 0 1
There is no preferred order for the columns of ~S. If had we written
0
1 1
1
0
B ~S ˆ @ 1 1
C
0A
0 0 1
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