14
1 Some Linear Algebra
(1) If T 1 and T 2 are unitary then also is T 1 ⊗ T 2 .
(2) If T 1 and T 2 are Hermitian then also is T 1 ⊗ T 2 .
(3) If T 1 and T 2 are projectors then T 1 ⊗ T 2 is also a projector.
(4) If T 1 and T 2 are positive operators then also is T 1 ⊗ T 2 .
Exercise 1.7.3 Show Proposition 1.7.2.
We are now interested to define tensor products of matrices since, from Theorem 1.2.2, linear transformations and matrices are equivalent (fixing the bases).
Definition 1.7.14 Let K a field. Assume that A = [a i j ] is an m × n and B = [b i j ]
is a r × s matrix both with entries in K . Then the Kronecker product A ⊗ B of A
and B is defined as
A ⊗ B =
⎡
⎢
⎢
⎢
⎣
a 11 B a 12 B · · · a 1n B
a 21 B a 22 B · · · a 2n B
. . .
. . .
. . .
. . .
a m1 B a m2 B · · · a mn B
⎤
⎥
⎥
⎥
⎦
.
Since vectors can be considered as column (or row) matrices we also have a tensor
product of vectors. As an example, if K = C and we consider the vectors v 1 ∈ C
2
and v 2 ∈ C
3 given, respectively, by
v 1 =
i
2
, v 2 =
⎡
⎣
1
2i
5
⎤
⎦ , then v 1 ⊗ v 2 =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
i
−2
5i
2
4i
10
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.
The Kronecker product satisfies nice properties.
Proposition 1.7.3 Let A and B be two matrices with complex entries. Then the
following hold:
(1) (A ⊗ B)
T
= A
T
⊗ B
T ;
(2) (A ⊗ B)
∗
= A
∗
⊗ B
∗ ;
(3) (A ⊗ B)
†
= A
†
⊗ B
† .
Exercise 1.7.4 Show Proposition 1.7.3.
1.8 Commutator
We now present the concept of commutators and anti-commutators of linear operators.