A New Canonical Form for Complex Symmetric Matrices

N. H. Scott


It is well known that every real symmetric matrix, and every (complex) hermitian matrix, is diagonalizable, i.e. orthogonally similar to a diagonal matrix. However, a complex symmetric matrix with repeated eigenvalues may fail to be diagonalizable. We present a block diagonal canonical form, in which each block is quasi-diagonal, to which every complex symmetric matrix is orthogonally similar. As far as applications are concerned, complex symmetric matrices, as opposed to hermitian matrices, play an important role in theories of wave propagation in continuous media (e.g. elasticity, thermoelasticity).