510 Chapter 18Matrices and linear transformations
But two matrices need not commute even if they are both square.
EXAMPLE 18.10Non-commuting matrices
If
then
but
so thatAB 1 ≠ 1 BA.
0 Exercise 39
Because of the possibility of non-commutation, it is essential that the correct order of
the factors in a product be observed. In the product AB, the matrix Bis premultiplied,
or multiplied from the left, by A; the matrix Ais postmultiplied, or multiplied from
the right, by B.
A quantity that plays an important role in quantum mechanics is the commutator
of the matrices Aand B,
[A, B] 1 = 1 AB 1 − 1 BA (18.32)
0 Exercises 40, 41
EXAMPLE 18.11The Pauli spin matrices
In quantum mechanics, electron spin is sometimes represented by the three Pauli
spin matrices, one for each cartesian component of the spin angular momentum,
(18.33)
SS Sxy zi
i
=
,=
−
,=
1
2
01
10
1
2
0
0
1
22
10
01
−
BA=
=
11
10
10
00
10
10
AB=
=
10
00
11
10
11
00
AB=
,=
10
00
11
10