Symmetric matrix: Difference between revisions

Revision as of 23:43, 2 March 2021

Revision as of 23:45, 2 March 2021

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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 a complex symmetric matrix <math>A</math> is diagonalizable it may be decomposed as

:<math>A = Q \Lambda Q^\textsf{T}</math>

where <math>Q</math> is a complexan 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>x</math> and <math>y</math> are eigenvectors corresponding to distinct eigenvalues <math>\lambda_1</math>, <math>\lambda_2</math>. Then

:<math>\lambda_1 \langle x, y \rangle = \langle Ax, y \rangle = \langle x, Ay \rangle = \lambda_2 \langle x, y \rangle.</math>