Symmetric matrix
A square matrix in which any two elements symmetrically positioned with respect to the main diagonal are equal to each other, that is, a matrix 
that is equal to its transpose:
 |
A real symmetric matrix of order 
has exactly 
real eigenvalues (counted with multiplicity). If 
is a symmetric matrix, then so are 
and 
, and if 
and 
are symmetric matrices of the same order, then 
is a symmetric matrix, while 
is symmetric if and only if 
.
Comments
Every square complex matrix is similar to a symmetric matrix. A real 
-matrix is symmetric if and only if the associated operator 
(with respect to the standard basis) is self-adjoint (with respect to the standard inner product). A polar decomposition factors a matrix 
into a product 
of a symmetric and an orthogonal matrix.
Let 
be a bilinear form on a vector space 
(cf. Bilinear mapping). Then the matrix of 
(with respect to the same basis in the two factors 
) is symmetric if and only if 
is a symmetric bilinear form, i.e. 
.
References
| [a1] | F.R. [F.R. Gantmakher] Gantmacher, "The theory of matrices" , 1 , Chelsea, reprint (1959–1960) pp. Vol. 1, Chapt. IX; Vol. 2, Chapt. XI (Translated from Russian) |
| [a2] | W. Noll, "Finite dimensional spaces" , M. Nijhoff (1987) pp. Sect. 2.7 |
Symmetric matrix. T.S. Pigolkina (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Symmetric_matrix&oldid=16470