Editing Symmetric matrix (section)

== Properties ==
===Basic properties===
* The sum and difference of two symmetric matrices is symmetric.
* This is not always true for the [[matrix multiplication|product]]: given symmetric matrices <math>A</math> and <math>B</math>, then <math>AB</math> is symmetric if and only if <math>A</math> and <math>B</math> [[commutativity|commute]], i.e., if <math>AB=BA</math>.
* For any integer <math>n</math>, <math>A^n</math> is symmetric if <math>A</math> is symmetric.
* If <math>A^{-1}</math> exists, it is symmetric if and only if <math>A</math> is symmetric.
* Rank of a symmetric matrix <math>A</math> is equal to the number of non-zero eigenvalues of <math>A</math>.

===Decomposition into symmetric and skew-symmetric===
Any square matrix can uniquely be written as sum of a symmetric and a skew-symmetric matrix. This decomposition is known as the Toeplitz decomposition. Let <math>\mbox{Mat}_n</math> denote the space of <math>n \times n</math> matrices. If <math>\mbox{Sym}_n</math> denotes the space of <math>n \times n</math> symmetric matrices and <math>\mbox{Skew}_n</math> the space of <math>n \times n</math> skew-symmetric matrices then <math>\mbox{Mat}_n = \mbox{Sym}_n + \mbox{Skew}_n</math> and <math>\mbox{Sym}_n \cap \mbox{Skew}_n = \{0\}</math>, i.e.
<math display="block">\mbox{Mat}_n = \mbox{Sym}_n \oplus \mbox{Skew}_n , </math>
where <math>\oplus</math> denotes the [[direct sum of modules|direct sum]]. Let <math>X \in \mbox{Mat}_n</math> then
<math display="block">X = \frac{1}{2}\left(X + X^\textsf{T}\right) + \frac{1}{2}\left(X - X^\textsf{T}\right).</math>

Notice that <math display="inline">\frac{1}{2}\left(X + X^\textsf{T}\right) \in \mbox{Sym}_n</math> and <math display="inline">\frac{1}{2} \left(X - X^\textsf{T}\right) \in \mathrm{Skew}_n</math>. This is true for every [[square matrix]] <math>X</math> with entries from any [[field (mathematics)|field]] whose [[characteristic (algebra)|characteristic]] is different from 2.

A symmetric <math>n \times n</math> matrix is determined by <math>\tfrac{1}{2}n(n+1)</math> scalars (the number of entries on or above the [[main diagonal]]). Similarly, a [[skew-symmetric matrix]] is determined by <math>\tfrac{1}{2}n(n-1)</math> scalars (the number of entries above the main diagonal).

=== Matrix congruent to a symmetric matrix ===
Any matrix [[matrix congruence|congruent]] to a symmetric matrix is again symmetric: if <math>X</math> is a symmetric matrix, then so is <math>A X A^{\mathrm T}</math> for any matrix <math>A</math>.

=== Symmetry implies normality ===
A (real-valued) symmetric matrix is necessarily a [[normal matrix]].

=== 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.

=== Complex symmetric matrices {{anchor|Complex}}===
A complex symmetric matrix can be 'diagonalized' using a [[unitary matrix]]: thus if <math>A</math> is a complex symmetric matrix, there is a unitary matrix <math>U</math> such that  <math>U A U^{\mathrm T}</math> is a real diagonal matrix with non-negative entries. This result is referred to as the '''Autonne–Takagi factorization'''. It was originally proved by [[Léon Autonne]] (1915) and [[Teiji Takagi]] (1925) and rediscovered with different proofs by several other mathematicians.<ref>{{harvnb|Horn|Johnson|2013|pp=263,278}}</ref><ref>See:
*{{citation|first=L.|last= Autonne|title= Sur les matrices hypohermitiennes et sur les matrices unitaires|journal= Ann. Univ. Lyon|volume= 38|year=1915|pages= 1–77|url=https://gallica.bnf.fr/ark:/12148/bpt6k69553b}}
*{{citation|first=T.|last= Takagi|title= On an algebraic problem related to an analytic theorem of Carathéodory and Fejér and on an allied theorem of Landau|journal= Jpn. J. Math.|volume= 1 |year=1925|pages= 83–93|doi= 10.4099/jjm1924.1.0_83|doi-access= free}}
*{{citation|title=Symplectic Geometry|first=Carl Ludwig|last= Siegel|journal= American Journal of Mathematics|volume= 65|issue=1 |year=1943|pages=1–86|jstor= 2371774|doi=10.2307/2371774|id=Lemma 1, page 12}}
*{{citation|first=L.-K.|last= Hua|title= On the theory of automorphic functions of a matrix variable I–geometric basis|journal= Amer. J. Math.|volume= 66 |issue= 3|year=1944|pages= 470–488|doi=10.2307/2371910|jstor= 2371910}}
*{{citation|first=I.|last= Schur|title= Ein Satz über quadratische Formen mit komplexen Koeffizienten|journal=Amer. J. Math. |volume=67 |issue= 4|year=1945|pages=472–480|doi=10.2307/2371974|jstor= 2371974}}
*{{citation|first1=R.|last1= Benedetti|first2=P.|last2= Cragnolini|title=On simultaneous diagonalization of one Hermitian and one symmetric form|journal= Linear Algebra Appl. |volume=57 |year=1984| pages=215–226|doi=10.1016/0024-3795(84)90189-7|doi-access=free}}
</ref> In fact, the matrix <math>B=A^{\dagger} A</math> is Hermitian and [[Definiteness of a matrix|positive semi-definite]], so there is a unitary matrix <math>V</math> such that <math>V^{\dagger} B V</math> is diagonal with non-negative real entries. Thus <math>C=V^{\mathrm T} A V</math> is complex symmetric with <math>C^{\dagger}C</math> real. Writing <math>C=X+iY</math> with <math>X</math> and <math>Y</math> real symmetric matrices,  <math>C^{\dagger}C=X^2+Y^2+i(XY-YX)</math>. Thus <math>XY=YX</math>. Since <math>X</math> and <math>Y</math> commute, there is a real orthogonal matrix <math>W</math> such that both <math>W X W^{\mathrm T}</math> and <math>W Y W^{\mathrm T}</math> are diagonal. Setting <math>U=W V^{\mathrm T}</math> (a unitary matrix), the matrix <math>UAU^{\mathrm T}</math> is complex diagonal. Pre-multiplying <math>U</math> by a suitable diagonal unitary matrix (which preserves unitarity of <math>U</math>), the diagonal entries of <math>UAU^{\mathrm T}</math> can be made to be real and non-negative as desired. To construct this matrix, we express the diagonal matrix as <math>UAU^\mathrm T = \operatorname{diag}(r_1 e^{i\theta_1},r_2 e^{i\theta_2}, \dots, r_n e^{i\theta_n})</math>. The matrix we seek is simply given by <math>D = \operatorname{diag}(e^{-i\theta_1/2},e^{-i\theta_2/2}, \dots, e^{-i\theta_n/2})</math>. Clearly <math>DUAU^\mathrm TD = \operatorname{diag}(r_1, r_2, \dots, r_n)</math> as desired, so we make the modification <math>U' = DU</math>. Since their squares are the eigenvalues of <math>A^{\dagger} A</math>, they coincide with the [[singular value]]s of <math>A</math>. (Note, about the eigen-decomposition of a complex symmetric matrix <math>A</math>, the Jordan normal form of <math>A</math> may not be diagonal, therefore <math>A</math> may not be diagonalized by any similarity transformation.)