Chapter 4
Computation of the Eigenvectors
In this chapter methods are discussed how to compute the eigenvectors $v_1^*, v_2^*, \dots, v_n^*$ of the synthesis matrix $S$ as defined in (2.3), given the eigenvalues $\sigma_1^*, \sigma_2^*, \dots, \sigma_n^*$ of $S$ as computed in Section 3.
The main goal is to compute a full set of orthonormal eigenvectors using only $O(n^2)$ operations (for $b \ll n$). In order to achieve this goal, explicit reorthogonalization may only be used on very small sets of eigenvectors. Two different approaches are possible.
Computing the eigenvectors directly using newly developed explicit formulas;
Performing inverse iteration applied to the matrix $\Sigma + UU$ (see Gansterer et al. [16]).
The latter approach in its standard form may fail to achieve the goal of applying only $O(n^2)$ operations in case reorthogonalization is required (Dhillon [10]), and therefore the former approach will be pursued in this report.
In this chapter it will be shown how an orthonormal basis for the eigenspace of $S$ can be constructed. Since eigenvectors corresponding to distinct eigenvalues are orthogonal (Horn and Johnson [19]) only an orthonormal basis of the eigenspace corresponding to one given eigenvalue $\sigma_i^*$ has to be constructed.
Three cases have to be distinguished.
Case I
In the first case assume that $\sigma_i^* \notin \lambda(\Sigma)$ and has an algebraic multiplicity $l$, $1 \le l \le b$. Since $\sigma_i^*$ is an eigenvalue of $S$, the matrix $\Sigma + UU - \sigma_i^* I_n$ is singular. This implies the existence of an $n$-vector $x \ne 0$ (a corresponding eigenvector), such that
Premultiplication with $(\Sigma - \sigma_i^* I_n)^{-1}$ (which exists because of the assumption $\sigma_i^* \notin \lambda(\Sigma)$) yields
It follows that
is a singular matrix. Similarly, premultiplication with $U (\Sigma - \sigma_i^* I_n)^{-1}$ yields