Symmetric matrix: Difference between revisions

Revision as of 21:59, 2 February 2021

Revision as of 23:43, 2 March 2021

Line 107:

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 complex symmetric matrix may be [[defective matrix|defective]] and thus not be diagonalizable. byIf similarity;a every realcomplex symmetric matrix $A$ is diagonalizable byit amay realbe orthogonaldecomposed similarity.as

Every complex symmetric matrix <math>A</math> can be diagonalized by unitary congruence

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

where <math>Q</math> is a [[unitarycomplex orthogonal matrix]]. If<math>Q AQ^\textsf{T} is= realI</math>, the matrixand <math>Q\Lambda</math> is a real [[orthogonaldiagonal matrix]], (the columns of whichthe are [[eigenvectors]]eigenvalues of <math>A</math>),. andIn the special case that <math>\LambdaA</math> is real andsymmetric, diagonalthen (having<math>Q</math> the [[eigenvalues]] ofand <math>A\Lambda</math> onare thealso diagonal)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>