Complex symmetric matrices
- 1 November 1969
- journal article
- research article
- Published by Cambridge University Press (CUP) in Journal of the Australian Mathematical Society
- Vol. 10 (3-4) , 341-354
- https://doi.org/10.1017/s1446788700007588
Abstract
It is well known that a real symmetric matrix can be diagonalised by an orthogonal transformation. This statement is not true, in general, for a symmetric matrix of complex elements. Such complex symmetric matrices arise naturally in the study of damped vibrations of linear systems. It is shown in this paper that a complex symmetric matrix can be diagonalised by a (complex) orthogonal transformation, when and only when each eigenspace of the matrix has an orthonormal basis; this implies that no eigenvectors of zero Euclidean length need be included in the basis. If the matrix cannot be diagonalised, then it has at least one invariant subspace which consists entirely of vectors of zero Euclidean length.Keywords
This publication has 1 reference indexed in Scilit:
- Über symmetrische, alternierende und orthogonale Normalformen von Matrizen.Journal für die reine und angewandte Mathematik (Crelles Journal), 1930