  Consider the optimal subspace expansion problem for the matrix eigenvalue
problem $Ax=\lambda x$: Which vector $w$ in the current subspace $\mathcal{V}$,
after multiplied by $A$, provides an optimal subspace expansion for
approximating a desired eigenvector $x$ in the sense that $x$ has the smallest
angle with the expanded subspace $\mathcal{V}_w=\mathcal{V}+{\rm span}\{Aw\}$,
i.e., $w_{opt}=\arg\max_{w\in\mathcal{V}}\cos\angle(\mathcal{V}_w,x)$? This
problem is important as many iterative methods construct nested subspaces that
successively expand $\mathcal{V}$ to $\mathcal{V}_w$. An expression of
$w_{opt}$ by Ye (Linear Algebra Appl., 428 (2008), pp. 911--918) for $A$
general, but it could not be exploited to construct a computable (nearly)
optimally expanded subspace. He turns to deriving a maximization
characterization of $\cos\angle(\mathcal{V}_w,x)$ for a {\em given} $w\in
\mathcal{V}$ when $A$ is Hermitian. We generalize Ye's maximization
characterization to the general case and find its maximizer. Our main
contributions consist of explicit expressions of $w_{opt}$, $(I-P_V)Aw_{opt}$
and the optimally expanded subspace $\mathcal{V}_{w_{opt}}$ for $A$ general,
where $P_V$ is the orthogonal projector onto $\mathcal{V}$. These results are
fully exploited to obtain computable optimally expanded subspaces within the
framework of the standard, harmonic, refined, and refined harmonic
Rayleigh--Ritz methods. We show how to efficiently implement the proposed
subspace expansion approaches. Numerical experiments demonstrate the
effectiveness of our computable optimal expansions.
