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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> {{Relevance inline|reason=not referred to in this section|date=December 2015}} 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=G.H. Golub, C.F. van Loan. | title=Matrix Computations | publisher=The Johns Hopkins University Press, Baltimore, London | year=1996}}</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
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