Mathematical Methods for Physicists: A concise introduction - Site Map

SYMMETRIC AND SKEW-SYMMETRIC MATRICES
so that
…AB† T ˆ B T A T
q:e:d:
Because of (3.15), even if ~ A ˆ ~A T and ~B ˆ ~B T , … ~ A ~B† T 6ˆ ~A ~B unless the matrices
commute.
Symmetric and skew-symmetric matrices
A square matrix ~A ˆ…a jk † is said to be symmetric if all its elements satisfy the
equations
a kj ˆ a jk ;
that is, ~A and its transpose are equal ~A ˆ ~A T . For example,
0
1
1 5 7
B
C
~A ˆ @ 5 3 4 A
7 4 0
…3:16†
is a third-order symmetric matrix: the elements of the ith row equal the elements
of ith column, for all i.
On the other hand, if the elements of ~A satisfy the equations
a kj ˆa jk ;
…3:17†
then ~A is said to be skew-symmetric, or antisymmetric. Thus, for a skew-symmetric
~A, its transpose equals minus ~A: ~A T ˆ~A.
Since the elements a jj along the principal diagonal satisfy the equations
a jj ˆa jj , it is evident that they must all vanish. For example,
0
1
0 2 5
B
C
~A ˆ @ 2 0 1A
5 1 0
is a skew-symmetric matrix.
Any real square matrix ~A may be expressed as the sum of a symmetric matrix ~R
and a skew-symmetric matrix ~S, where
~R ˆ 1
2 … ~A ‡ ~A T † and ~S ˆ 1
2 … ~A ~A T †: …3:18†
Example 3.7
The matrix
~A ˆ 2 3
5 1
may be written in the form ~A ˆ ~R ‡ ~S, where
~R ˆ 1
2 … ~A ‡ ~A T †ˆ 2 4
4 1
~S ˆ 1
2 … ~A ~A T †ˆ 0 1
1 0
:
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