Mathematical Methods for Physicists: A concise introduction - Site Map

THE COMMUTATOR
The commutator
Even if matrices ~A and ~B are both square matrices of order n, the products ~A ~B
and ~B ~A, although both square matrices of order n, are in general quite di€erent,
since their individual elements are formed di€erently. For example,
1 2 1 0
1 3 1 2
ˆ 3 4
4 6
but
1 0 1 2
1 2 1 3
ˆ 1 2
3 8
The di€erence between the two products ~A ~B and ~B ~A is known as the commutator
of ~A and ~B and is denoted by
It is obvious that
:
‰ ~A; ~BŠ ˆ ~A ~B ~B ~A: …3:12†
‰ ~B; ~AŠ ˆ‰~A; ~BŠ: …3:13†
If two square matrices ~A and ~B are very carefully chosen, it is possible to make the
product identical. That is ~A ~B ˆ ~B ~A. Two such matrices are said to commute with
each other. Commuting matrices play an important role in quantum mechanics.
If ~A commutes with ~B and ~B commutes with ~C, it does not necessarily follow
that ~ A commutes with ~ C.
Powers of a matrix
If n is a positive integer and ~A is a square matrix, then ~A 2 ˆ ~A ~A, ~A 3 ˆ ~A ~A ~A, and
in general, ~A n ˆ ~A ~A ~A (n times). In particular, ~A 0 ˆ ~I.
Functions of matrices
As we de®ne and study various functions of a variable in algebra, it is possible to
de®ne and evaluate functions of matrices. We shall brie¯y discuss the following
functions of matrices in this section: integral powers and exponential.
A simple example of integral powers of a matrix is polynomials such as
f … ~A† ˆ ~A 2 ‡ 3 ~A 5 :
Note that a matrix can be multiplied by itself if and only if it is a square matrix.
Thus ~A here is a square matrix and we denote the product ~A ~A as ~A 2 . More fancy
examples can be obtained by taking series, such as
~S ˆ X1
kˆ0
a k
~ A k ;
where a k are scalar coecients. Of course, the sum has no meaning if it does not
converge. The convergence of the matrix series means every matrix element of the
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