Complex symmetric matrices

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.

On powers of general tridiagonal matrices

“…One of the earliest articles is [6], which states that complex symmetric matrices can be diagonalized by a complex(orthogonal) transformation if and only if each eigenspace of the matrix has an orthonormal basis, which means that no eigenvectors have zero Euclidian norm are included in the basis, such eigenvectors are called quasi-null vectors, they are nonzero vectors but with zero norm, such as the vector 1 + i . [14] discusses this for a special type of complex symmetric matrices, namely the matrices which have positive definite real and imaginary parts.…”

“…Consequently, a quite large number of publications that address this subject appeared in recent years. Some of these publications discuss inversion of these matrices in general, or specially structured types of them [1,2,4,5,6,7,10,11], some other publications discuss powers of such specially structured matrices [8,9,12,13,14,15], and some discuss both inversion and powers of these matrices [3].…”

A two-level domain decomposition method for the iterative solution of high frequency exterior Helmholtz problems

“…In this work, we choose to solve the symmetric complex interface problem (19) by the CR method equipped with the "symmetric" rather than Hermitian inner product [26,28]. More specifically, we adopt the computer implementation of CR described in [27].…”

“…This requires transforming each iterate λ k into the iterateλ k as described in Eqs. (24)(25)(26)(27)(28). This in turn requires solving at each iteration the second-level FETI problem (25).…”

Integrability conditions of geodesic flow on homogeneous Monge manifolds

“…Still, complex symmetric matrices display a rich structure presented in Craven [4]. We will use in particular that a diagonalizable complex symmetric matrix is always diagonalizable in an 'orthonormal' basis (for the pseudo-norm x 2 = x 2 1 + · · · + x 2 n ).…”