A symmetric matrix is a square matrix that satisfies
 |
(1)
|
where 
denotes the transpose, so 
. This also implies
 |
(2)
|
where 
is the identity matrix. For example,
![A=[4 1; 1 -2]](/images/equations/SymmetricMatrix/NumberedEquation3.svg) |
(3)
|
is a symmetric matrix. Hermitian matrices are a useful generalization of symmetric matrices for complex matrices.
A matrix that is not symmetric is said to be an asymmetric matrix, not to be confused with an antisymmetric matrix.
A matrix 
can be tested to see if it is symmetric in the Wolfram
Language using SymmetricMatrixQ[m].
Written explicitly, the elements of a symmetric matrix 
have the form
![[a_(11) a_(12) ... a_(1n); a_(12) a_(22) ... a_(2n); | | ... |; a_(1n) a_(2n) ... a_(nn)].](/images/equations/SymmetricMatrix/NumberedEquation4.svg) |
(4)
|
The symmetric part of any matrix may be obtained from
 |
(5)
|
A matrix 
is symmetric if it can be expressed in the form
 |
(6)
|
where 
is an orthogonal matrix and 
is a diagonal matrix. This
is equivalent to the matrix equation
 |
(7)
|
which is equivalent to
 |
(8)
|
for all 
,
where 
.
Therefore, the diagonal elements of 
are the eigenvalues of 
, and the columns of 
are the corresponding eigenvectors.
The numbers of symmetric matrices of order 
on 
symbols are 
, 
, 
, 
, ..., 
. Therefore, for (0,1)-matrices,
the numbers of distinct symmetric matrices of orders 
, 2, ... are 2, 8, 64, 1024, ... (OEIS A006125).
