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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<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-linklink1=Gene H. Golub |lastlast1=Golub |firstfirst1=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

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The transpose of a symmetrizable matrix is symmetrizable, since <math>A^{\mathrm T}=(DS)^{\mathrm T}=SD=D^{-1}(DSD)</math> and <math>DSD</math> is symmetric. A matrix <math>A=(a_{ij})</math> is symmetrizable if and only if the following conditions are met:

# <math>a_{ij} = 0</math> implies <math>a_{ji} = 0</math> for all <math>1 \le i \le j \le n.</math>

# <math>a_{i_1 i_2} a_{i_2 i_3} \dots a_{i_k i_1} = a_{i_2 ia_i_1} a_{i_3 i_2} \dots a_{i_1 i_k}</math> for any finite sequence <math>\left(i_1, i_2, \dots, i_k\right).</math>

== See also ==

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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}}

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[[Category:Matrices (mathematics)]]