A symmetric matrix is a Square Matrix which satisfies
where
denotes the
Transpose, so
. This also implies

(1) 
where I is the Identity Matrix. Written explicitly,

(2) 
The symmetric part of any Matrix may be obtained from

(3) 
A Matrix A is symmetric if it can be expressed in the form

(4) 
where
is an Orthogonal Matrix and
is a Diagonal Matrix. This is equivalent to the
Matrix equation

(5) 
which is equivalent to

(6) 
for all , where
. Therefore, the diagonal elements of
are the
Eigenvalues of
, and the columns of
are the corresponding Eigenvectors.
See also Antisymmetric Matrix, Skew Symmetric Matrix
References
Nash, J. C. ``Real Symmetric Matrices.''
Ch. 10 in Compact Numerical Methods for Computers: Linear Algebra
and Function Minimisation, 2nd ed. Bristol, England: Adam Hilger, pp. 119134, 1990.
© 19969 Eric W. Weisstein
19990526