Power method

Iterate $q^{(k+1)}=A{q}^{(k)}/\Vert A{q}^{(k)}\Vert$. With a unique dominant eigenvalue separated by a spectral gap and a starting vector that is nonorthogonal to its eigenvector, the directions converge at the spectral-ratio rate.

For a real symmetric matrix with $|{\lambda }_1|>|{\lambda }_2|$ and a starting vector having nonzero component in the leading eigendirection, power-method direction errors arewhereas the Rayleigh quotient error isThe improvement occurs because orthogonality cancels terms linear in the direction error.

The active spectrum consists of eigenvalues whose eigenspaces have nonzero projection of the starting vector. The power method converges toward the eigenvalue of largest modulus in this active spectrum; an exactly absent dominant component is never created in exact arithmetic.

Inverse iteration repeatedly solves $(A-sI)y_{k+1}=q_k$ and normalizes. It is the power method for $(A-sI{)}^{-1}$ and converges toward an eigenvector whose eigenvalue is closest to the shift $s$.

If $A$ is real symmetric, $s$ is not an eigenvalue, one eigenvalue ${\lambda }_{\ast }$ is uniquely closest to $s$, and the initial vector has a nonzero component in its eigenspace, fixed-shift inverse iteration converges to that eigenspace. Its asymptotic direction-error ratio iswhere ${\lambda }_{\mathrm{next}}$ is second closest to $s$.

If the closest eigenvalue ${\lambda }_{\ast }$ lies above the shift, the dominant eigenvalue $({\lambda }_{\ast }-s{)}^{-1}$ of the inverse is positive and normalized iterates approach one fixed sign of its eigenvector. If ${\lambda }_{\ast }<s$, that inverse eigenvalue is negative, so the direction converges but consecutive normalized vectors alternate signs. At a shift exactly midway between two simple eigenvalues, the two inverse eigenvalues have equal modulus and opposite signs, so generic iterates alternate between two mixtures instead of selecting one eigenvector.

Subspace iteration repeatedly applies a matrix to several independent vectors and orthonormalizes them, converging to a dominant invariant subspace when a spectral gap separates the selected eigenvalue cluster.

A dominant invariant subspace is spanned by eigenvectors whose eigenvalue moduli exceed those outside the subspace.

When several eigenvalues share the largest modulus, normalized power iterates need not converge individually, but every limit lies in their joint invariant subspace.