Symmetric matrix: Difference between revisions

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* 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>.

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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|authorfirst=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>{{cite bookharvnb|first1=R.A.|last1=Horn|first2=C.R.|last2=Johnson|title=Matrix analysis |year=2013 | editionpp=2nd | publisher = Cambridge University Press | mr = 2978290|at=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}}

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<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=G.Gene H. Golub, C|last1=Golub |first1=G.H. |author2-link=Charles F. Van Loan |last2=van Loan |first2=C.F. | title=Matrix Computations | publisher=The Johns Hopkins University Press, Baltimore, London | 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

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== Hessian ==

Symmetric <math>n \times n</math> matrices of real functions appear as the [[Hessian matrix|Hessians]] of twice differentiable functions of <math>n</math> real variables (the continuity of the second derivative is not needed, despite common belief to the opposite<ref>{{Cite book |last=Dieudonné |first=Jean A. |title=Foundations of Modern Analysis |publisher=Academic Press |year=1969 |isbn=978-1443724265 |edition=Enlarged and Corrected printing |pageschapter=Theorem (8.12.2), p.|page=180 180|isbn=0-12-215550-5 |languageoclc=en576465}}</ref>).

Every [[quadratic form]] <math>q</math> on <math>\mathbb{R}^n</math> can be uniquely written in the form <math>q(\mathbf{x}) = \mathbf{x}^\textsf{T} A \mathbf{x}</math> with a symmetric <math>n \times n</math> matrix <math>A</math>. Because of the above spectral theorem, one can then say that every quadratic form, up to the choice of an orthonormal basis of <math>\R^n</math>, "looks like"

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== References ==

{{refbegin}}

*{{citation|lastlast1=Horn|firstfirst1= Roger A.|last2= Johnson|first2= Charles R.|title= Matrix analysis|edition=2nd| publisher=Cambridge University Press|year= 2013|isbn= 978-0-521-54823-6}}

{{refend}}

== External links ==

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[[Category:Matrices (mathematics)]]