Symmetric matrix: Difference between revisions

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== Properties ==

One of the basic theorems concerning such matrices is the finite-dimensional [[spectral theorem]], which says that any symmetric matrix whose entries are [[real number|real]] can be diagonalized by an [[orthogonal matrix]]. More Thisexplicitely: isto every symmetric real matrix ''A'' there exists a specialreal caseorthogonal ofmatrix ''Q'' such that ''D'' = ''Q''^T''AQ'' is a [[Hermitiandiagonal matrix]].

Every real symmetric matrix is [[Hermitian matrix|Hermitian]], and thus all its eigenvalues are real. (In fact, they are the entries in the above diagonal matrix ''D'', and therefore ''D'' is uniquely determined by ''A'', [[up to]] the order of its entries.) Essentially, the property of symmetry of real matrices corresponds to the property of being Hermitian for complex matrices.

See also [[skew-symmetric matrix|skew-symmetric]] (or antisymmetric) matrix.

Using the [[Jordan normal form]], one can prove that every square real matrix can be written as the product of two symmetric matrices.

== See also ==

Other types of [[symmetry]] or pattern in square matrices have special names: see for example:

See also *[[skew-symmetric matrix|skew-symmetric]] (or antisymmetric) matrix.

*[[circulant matrix]]

*[[Hankel matrix]]