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. 119-134, 1990.
© 1996-9 Eric W. Weisstein
1999-05-26