19.7 Exercises 557
Repeat Exercise 25 for:
- of Exercise 4 27. of Exercise 8
- of Exercise 9
Section 19.5
Express in matrix form:
- 5 x
2
1 − 12 xy 1 − 13 y
2
- 4 xy 31. 3 x
2
1 − 14 xy 1 + 12 xz 1 − 16 yz 1 + 1 y
2
1 − 12 z
2
Transform the following quadratic forms into canonical form:
- 33.ax
2
1 + 12 bxy 1 + 1 ay
2
- 3 x
1
2
1 + 12 x
1
x
2
1 + 12 x
1
x
4
1 + 13 x
2
2
1 + 12 x
2
x
3
1 + 13 x
3
2
1 + 12 x
3
x
4
1 + 13 x
4
2
35.Derive equations (19.44) for the components of the inertia tensor.
[Hint: Expand equation (16.60),l 1 = 1 mr
2
y 1 − 1 m(r 1
·1 y)r, for the angular momentum in
terms of components.]
Section 19.6
Find the complex conjugate and Hermitian conjugate of the following matrices
39.If and , find (i) a
†
a (ii)b
†
b (iii)a
†
b (iv)b
†
a
40.Which of the matrices in Exercise 36–38 are Hermitian?
- (i)Show that is unitary.
(ii)Confirm that both the columns and the rows of Aform unitary systems of vectors.
42.Repeat Exercise 41 for
A=−
−
ii i
iii
ii
32 6
326
30 26
.
A=
−−
12 2
212
i
i
b=
2
0
3
i
a=
−
i
i
1
00
0
00
−
−
i
ii
i
2
1
i
−i
12
3
+−
+−
ii
ii
763 13
1
2
12 2
2
xxxx++
A=
030
303
030
A=
120
210
021
A=
22
13