46 1 Mathematical Physics
Fig. 1.10bReflection about a
line passing through origin at
45 ◦
Fig. 1.10cElongating a
vector in the same direction
Here the magnitude becomes double without changing its orientation.DX=
⎛
⎜
⎜
⎝
√
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 matrixDis a rotation matrix which rotates the vector through 30◦about
thez-axis,. Fig.1.10d.Fig. 1.10dRotation of a
vector through 30◦
1.30 The matrixA=
⎛
⎝
6 − 22
− 23 − 1
2 − 13
⎞
⎠
The characteristic equation is|A−λI|=∣ ∣ ∣ ∣ ∣ ∣
6 −λ − 22
− 23 −λ − 1
2 − 13 −λ∣ ∣ ∣ ∣ ∣ ∣
= 0
This gives−λ^3 + 12 λ^2 − 36 λ+ 32 = 0
or (λ−2)(λ−2)(λ−8)= 0
The characteristic roots (eigen values) are
λ 1 = 2 ,λ 2 =2 andλ 3 = 8