16
1 Some Linear Algebra
and
T 1 T 2 = −T 2 T 1 .
(1.2)
Applying T
−1
1 to the left in Eq. (1.1) it follows that
T 2 = T
−1
1 T 2 T 1 .
(1.3)
Replacing Eq. (1.2) in Eq. (1.3) it follows that T 2 must be the zero operator.
Exercise 1.8.2 Let T 1 , T 2 : V −→ V be linear operators. Show that
(a) [T 1 , T 2 ]
†
= [T
†
2 , T
†
1 ];
(b) [T 1 , T 2 ] = −[T 2 , T 1 ];
(c) i[T 1 , T 2 ] is Hermitian;
(d) Is it true that if T 1 and T 2 are unitary operators then so is [T 1 , T 2 ]? What about
this assertion with respect to anti-commutator?
1 Some Linear Algebra
and
T 1 T 2 = −T 2 T 1 .
(1.2)
Applying T
−1
1 to the left in Eq. (1.1) it follows that
T 2 = T
−1
1 T 2 T 1 .
(1.3)
Replacing Eq. (1.2) in Eq. (1.3) it follows that T 2 must be the zero operator.
Exercise 1.8.2 Let T 1 , T 2 : V −→ V be linear operators. Show that
(a) [T 1 , T 2 ]
†
= [T
†
2 , T
†
1 ];
(b) [T 1 , T 2 ] = −[T 2 , T 1 ];
(c) i[T 1 , T 2 ] is Hermitian;
(d) Is it true that if T 1 and T 2 are unitary operators then so is [T 1 , T 2 ]? What about
this assertion with respect to anti-commutator?