While the complex symmetry of has no effect on the eigenvalues of , this particular structure can be exploited to halve the work and storage requirements of the general non-Hermitian Lanczos method described in §7.8. Indeed, while the non-Hermitian Lanczos method involves one matrix-vector product with and one with at each iteration, the complex symmetric Lanczos method only requires one matrix-vector product with at each iteration.

After iterations, the complex symmetric Lanczos method
has generated Lanczos vectors,

Note that, in view of the eigendecomposition (7.91) of diagonalizable complex symmetric matrices , the complex orthogonality (7.95) of the Lanczos vectors is natural.

The complex symmetric Lanczos algorithm computes the
vectors (7.94) by means of three-term recurrences
that can be summarized as follows:

is a complex symmetric tridiagonal matrix whose entries are the coefficients of the three-term recurrences. The vector is the candidate for the next Lanczos vector, . It is constructed so that the orthogonality condition

is satisfied, and it only remains to be normalized so that . However, it cannot be excluded that

If (7.99) occurs, then a next vector cannot be obtained by simply normalizing , as it would require division by zero. Therefore, (7.99) is called a

After iterations of the complex symmetric Lanczos algorithm,
approximate eigensolutions for the complex symmetric
eigenvalue problem (7.88) are obtained by
computing eigensolutions of ,

Indeed, the relation follows by multiplying (7.96) from the left by and by using the orthogonality relations (7.95) and (7.98). Of course, in the complex symmetric Lanczos algorithm, the matrix is not computed via the relation (7.101). Instead, the symmetric tridiagonal structure in the definition (7.97) is exploited and only the diagonal and subdiagonal entries of are explicitly generated.

It should be pointed out that is complex orthogonal, but not unitary, which may have effects for the numerical stability.