Power method ($q^{(k+1)}=A{q}^{(k)}/\Vert A{q}^{(k)}\Vert$)

Iterate $q^{(k+1)}=A{q}^{(k)}/\Vert A{q}^{(k)}\Vert$. For a self-adjoint operator with a unique dominant eigenvalue separated in modulus from the others, a starting vector that is nonorthogonal to its dominant eigenvector gives convergence of the eigendirection at the spectral-ratio rate. For a general diagonalizable matrix, the condition is a nonzero coefficient in the dominant component of its eigenbasis, detected by the corresponding left eigenvector; an ordinary dot product with the right eigenvector does not generally detect that coefficient.

The power method is power iteration; its convergence depends on the dominant spectral component of the starting vector.

For example, $B=\left(\begin{array}{cc}2& 1\\ 0& 1\end{array}\right)$ has dominant right eigenvector $(1,0{)}^T$ and a dominant left eigenvector $(1,1{)}^T$. The starting vector $q^{(0)}=(-1,1{)}^T/\sqrt{2}$ has nonzero dot product with the right eigenvector, but $B{q}^{(0)}=q^{(0)}$: it stays forever in the eigenspace for the smaller eigenvalue $1$. Its coefficient along the dominant direction is zero, as the left eigenvector detects. An orthonormal eigenbasis makes the two tests agree in the self-adjoint case.

When two independent iterates span an invariant two-dimensional subspace, their recurrence gives the characteristic polynomial ${\lambda }^2-s\lambda +p$. For distinct roots ${\lambda }_1,{\lambda }_2$, corresponding eigenvectors are $y_{k+1}-{\lambda }_2{y}_k$ and $y_{k+1}-{\lambda }_1{y}_k$. For a merely approximately dominant subspace, the same computation gives approximate eigenpairs. A single orbit cannot separate a repeated eigenvalue's entire eigenspace when its projection is only one direction.

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.

For a diagonalizable matrix, the active spectrum consists of eigenvalues whose eigenspaces have nonzero components of the starting vector in its spectral decomposition. These components are orthogonal projections for a self-adjoint operator, but need not be orthogonal projections for a general diagonalizable matrix. With a unique largest modulus in this active spectrum, the power method selects its eigendirection; 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 $s$.

Suppose a diagonalizable matrix has a simple eigenvalue ${\lambda }_{\ast }$ uniquely closest to a shift $\mu \notin \mathrm{spec}(A)$. If the starting vector has a nonzero coefficient $c_{\ast }$ in the target eigenbasis direction, fixed-shift inverse iteration converges to that eigenvector line. In fact, the eigenvalues of $(A-\mu I{)}^{-1}$ are $({\lambda }_j-\mu {)}^{-1}$, so after $k$ inverse applications, division by $c_{\ast }({\lambda }_{\ast }-\mu {)}^{-k}$ leaves the target eigenvector plusThis proves direction error $O(q_{\mu }^k)$ in Big O notation. The multiplicative constant depends on the eigenbasis and starting coefficients, and no uniform reduction at the first step follows for a nonsymmetric matrix. Over the real numbers, ${\lambda }_{\ast }<\mu$ produces alternating signs. A nonzero ordinary inner product with a right eigenvector is not generally the same as a nonzero coefficient in the eigenbasis; the latter is the required condition.

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.