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## Properties of Complex Symmetric Matrices

Complex symmetry is a purely algebraic property, and it has no effect on the spectrum of the matrix. Indeed, for any given set of numbers, (195)

there exists a complex symmetric matrix whose eigenvalues are just the prescribed numbers (7.89); see, e.g., [233, Theorem 4.4.9].

A complex symmetric matrix may not even be diagonalizable. For example, consider the complex symmetric matrix (196)

The only eigenvalue of this matrix is , with algebraic multiplicity but geometric multiplicity . In fact, the Jordan normal form of is as follows: Thus, is not diagonalizable.

If a complex symmetric matrix is diagonalizable, then it has an eigendecomposition that reflects the complex symmetry; see, e.g., [233, Theorem 4.4.13]. More precisely, a complex symmetric matrix is diagonalizable if and only if its eigenvector matrix, , can be chosen such that (197)

A matrix with columns that satisfies is called complex orthogonal. The complex orthogonality of in (7.91) reflects the complex symmetry of .

We remark that the eigendecomposition (7.91) is the suitable adaptation of the corresponding decomposition for Hermitian matrices. Recall that for any matrix , the eigenvector matrix can always be chosen to be unitary: (198)

The unitariness of in (7.92) reflects the fact that is Hermitian.

The reason why an eigendecomposition (7.91) does not always exist is that there are complex vectors with (199)

Indeed, suppose has an eigenvalue with a one-dimensional eigenspace and the vector spanning that space satisfies (7.93). Then one of the columns of any eigenvector matrix of would be of the form , where is a scalar. Then, by (7.93), , while the complex orthogonality condition, , in (7.91) would imply . Note that for example (7.90), the vector spans the one-dimensional eigenspace associated with and it satisfies (7.93).     Next: Properties of the Algorithm Up: Lanczos Method for Complex Previous: Lanczos Method for Complex   Contents   Index
Susan Blackford 2000-11-20