Editing Symmetric matrix (section)

== Decomposition ==
Using the [[Jordan normal form]], one can prove that every square real matrix can be written as a product of two real symmetric matrices, and every square complex matrix can be written as a product of two complex symmetric matrices.<ref>{{cite journal | first=A. J.|last= Bosch | title=The factorization of a square matrix into two symmetric matrices | journal=[[American Mathematical Monthly]] | year=1986 | volume=93 | pages=462–464 | doi=10.2307/2323471 | issue=6 | jstor=2323471}}</ref>

Every real [[non-singular matrix]] can be uniquely factored as the product of an [[orthogonal matrix]] and a symmetric [[positive definite matrix]], which is called a [[polar decomposition]]. Singular matrices can also be factored, but not uniquely.

[[Cholesky decomposition]] states that every real positive-definite symmetric matrix <math>A</math> is a product of a lower-triangular matrix <math>L</math> and its transpose,
<math display="block">A = LL^\textsf{T}.</math>

If the matrix is symmetric indefinite, it may be still decomposed as <math>PAP^\textsf{T} = LDL^\textsf{T}</math> where <math>P</math> is a permutation matrix (arising from the need to [[pivot element|pivot]]), <math>L</math> a lower unit triangular matrix, and <math>D</math> is a direct sum of symmetric <math>1 \times 1</math> and <math>2 \times 2</math> blocks, which is called Bunch–Kaufman decomposition <ref>{{cite book |author-link1=Gene H. Golub |last1=Golub |first1=G.H. |author2-link=Charles F. Van Loan |last2=van Loan |first2=C.F. | title=Matrix Computations | publisher=Johns Hopkins University Press| year=1996 |isbn=0-8018-5413-X |oclc=34515797 }}</ref>

A general (complex) symmetric matrix may be [[defective matrix|defective]] and thus not be [[diagonalizable]]. If <math>A</math> is diagonalizable it may be decomposed as
<math display="block">A = Q \Lambda Q^\textsf{T}</math>
where <math>Q</math> is an orthogonal matrix <math>Q Q^\textsf{T} = I</math>, and <math>\Lambda</math> is a diagonal matrix of the eigenvalues of <math>A</math>. In the special case that <math>A</math> is real symmetric, then <math>Q</math> and <math>\Lambda</math> are also real. To see orthogonality, suppose <math>\mathbf x</math> and <math>\mathbf y</math> are eigenvectors corresponding to distinct eigenvalues <math>\lambda_1</math>, <math>\lambda_2</math>. Then
<math display="block">\lambda_1 \langle \mathbf x, \mathbf y \rangle = \langle A \mathbf x, \mathbf y \rangle = \langle \mathbf x, A \mathbf y \rangle = \lambda_2 \langle \mathbf x, \mathbf y \rangle.</math>

Since <math>\lambda_1</math> and <math>\lambda_2</math> are distinct, we have <math>\langle \mathbf x, \mathbf y \rangle = 0</math>.