46
1 Mathematical Physics
Fig. 1.10b Reflection about a
line passing through origin at
45
◦
Fig. 1.10c Elongating a
vector in the same direction
Here the magnitude becomes double without changing its orientation.
D X =
⎛
⎜
⎜
⎝
√
3
2
1
2
−
1
2
√
3
2
⎞
⎟
⎟
⎠
x
y
=
cos 30
◦ sin 30
◦
− sin 30
◦ cos 30
◦
x
y
=
⎛
⎜
⎜
⎝
√
3
2
x +
y
2
−
x
2
+
√
3
2
y
⎞
⎟
⎟
⎠
The matrix D is a rotation matrix which rotates the vector through 30
◦ about
the z-axis,. Fig.1.10d.
Fig. 1.10d Rotation of a
vector through 30
◦
1.30 The matrix A =
⎛
⎝
6 −2 2
−2 3 −1
2 −1 3
⎞
⎠
The characteristic equation is
|A − λI | =
6 − λ −2
2
−2 3 − λ −1
2 −1 3 − λ
= 0
This gives −λ
3
+ 12λ
2
− 36λ + 32 = 0
or (λ − 2)(λ − 2)(λ − 8) = 0
The characteristic roots (eigen values) are
λ 1 = 2, λ 2 = 2 and λ 3 = 8
1 Mathematical Physics
Fig. 1.10b Reflection about a
line passing through origin at
45
◦
Fig. 1.10c Elongating a
vector in the same direction
Here the magnitude becomes double without changing its orientation.
D X =
⎛
⎜
⎜
⎝
√
3
2
1
2
−
1
2
√
3
2
⎞
⎟
⎟
⎠
x
y
=
cos 30
◦ sin 30
◦
− sin 30
◦ cos 30
◦
x
y
=
⎛
⎜
⎜
⎝
√
3
2
x +
y
2
−
x
2
+
√
3
2
y
⎞
⎟
⎟
⎠
The matrix D is a rotation matrix which rotates the vector through 30
◦ about
the z-axis,. Fig.1.10d.
Fig. 1.10d Rotation of a
vector through 30
◦
1.30 The matrix A =
⎛
⎝
6 −2 2
−2 3 −1
2 −1 3
⎞
⎠
The characteristic equation is
|A − λI | =
6 − λ −2
2
−2 3 − λ −1
2 −1 3 − λ
= 0
This gives −λ
3
+ 12λ
2
− 36λ + 32 = 0
or (λ − 2)(λ − 2)(λ − 8) = 0
The characteristic roots (eigen values) are
λ 1 = 2, λ 2 = 2 and λ 3 = 8