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:
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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. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Symmetric_matrix&oldid=43512