Symmetric matrix: Difference between revisions

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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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== 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}}