Editing Symmetric matrix (section)

=== Real symmetric matrices ===
<!--If A is a skew-symmetric matrix, then ''iA'' (where ''i'' is an [[imaginary unit]]) is symmetric.-->
Denote by <math>\langle \cdot,\cdot \rangle</math> the standard [[inner product]] on <math>\mathbb{R}^n</math>. The real <math>n \times n</math> matrix <math>A</math> is symmetric if and only if
<math display="block">\langle Ax, y \rangle = \langle x, Ay \rangle \quad \forall x, y \in \mathbb{R}^n.</math>

Since this definition is independent of the choice of [[basis (linear algebra)|basis]], symmetry is a property that depends only on the [[linear operator]] A and a choice of [[inner product]]. This characterization of symmetry is useful, for example, in [[differential geometry]], for each [[tangent space]] to a [[manifold]] may be endowed with an inner product, giving rise to what is called a [[Riemannian manifold]]. Another area where this formulation is used is in [[Hilbert space]]s.

The finite-dimensional [[spectral theorem]] says that any symmetric matrix whose entries are [[real number|real]] can be [[diagonal matrix|diagonalized]] by an [[orthogonal matrix]]. More explicitly: For every real symmetric matrix <math>A</math> there exists a real orthogonal matrix <math>Q</math> such that <math>D = Q^{\mathrm T} A Q</math> is a [[diagonal matrix]]. Every real symmetric matrix is thus, [[up to]] choice of an [[orthonormal basis]], a diagonal matrix.

If <math>A</math> and <math>B</math> are <math>n \times n</math> real symmetric matrices that commute, then they can be simultaneously diagonalized by an orthogonal matrix:<ref>{{Cite book|first=Richard |last=Bellman|title=Introduction to Matrix Analysis |language= en|edition=2nd|publisher=SIAM|year=1997|isbn=08-9871-399-4}}</ref> there exists a basis of <math>\mathbb{R}^n</math> such that every element of the basis is an [[eigenvector]] for both <math>A</math> and <math>B</math>.

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