Computing Selected Eigenvalues

Shifted Inverse Iteration

Shifted inverse iteration $targets the eigenvalue nearest a chosen shift$ $\mu$ . We first study a real symmetric matrix $A \in \mathbb{R}^{n \times n}$ . All its eigenvalues are real, and its eigenvectors can be chosen as an orthonormal basis.

Choose $\mu$ that is not itself an eigenvalue and a unit starting vector $q_0$ with a nonzero component in the desired eigenvector direction. Factor $A-\mu I$ once, then reuse that factorization.

Assume the targeted eigenvalue is $uniquely closest to the shift$ . Equally close eigenvalues do not produce a unique dominant transformed magnitude. Convergence means convergence of the eigendirection. A negative transformed eigenvalue can alternate the vector sign.

One Iteration

For $k=0,1,2,\ldots$ , perform:

(A-\mu I)y_{k+1}=q_k

The Rayleigh quotient $\rho_{k+1}$ estimates the eigenvalue. A reliable stopping rule $checks the residual$ and accounts for the scale of the problem:

\|r_{k+1}\|_2 \leq \tau\bigl(\|A\|_2+|\rho_{k+1}|\bigr)

For an eigenpair satisfying $Av_i=\lambda_i v_i$ , the shifted inverse acts as follows.

(A-\mu I)^{-1}v_i=\frac{1}{\lambda_i-\mu}v_i

This relation shows that the eigenvalue closest to $\mu$ produces the largest transformed magnitude. In implementation, we solve the linear system without forming $(A-\mu I)^{-1}$ explicitly.

Power Iteration

Power iteration targets the eigenvalue of $largest magnitude$ . Starting from a unit vector $q_0$ , repeat:

y_{k+1}=Aq_k

For a symmetric matrix, convergence to the dominant eigenvector requires the following spectral gap.

|\lambda_1|>|\lambda_2|

The starting vector must also have a nonzero component in the direction of the eigenvector for $\lambda_1$ . The asymptotic convergence factor is approximately $|\lambda_2/\lambda_1|$ . Equal or tightly clustered dominant magnitudes can slow or prevent single-vector convergence.

The Eigenvalue Each Method Finds

The word “smallest” is ambiguous, so the target should always be stated precisely.

Method	Target
Power iteration on $A$	Eigenvalue with largest absolute value
Inverse iteration with $\mu=0$ and invertible $A$	Nonzero eigenvalue with smallest absolute value
Shifted inverse iteration	Eigenvalue closest to $\mu$
Inverse iteration on a positive-definite matrix	Smallest algebraic eigenvalue, which is also smallest in absolute value

For an indefinite matrix, the eigenvalue with $smallest absolute value need not$ be the most negative eigenvalue. This distinction prevents a common interpretation error.

Worked Shifted Inverse Example

Consider

A=\begin{pmatrix}2&0\\0&5\end{pmatrix}

The shifted system for the first iteration gives

y_1=(A-I)^{-1}q_0=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\1/4\end{pmatrix}

Because $2$ is closer to the shift $1$ than $5$ , further iterations drive the $second component toward zero$ and $\rho_k$ toward $2$ .

Check the first residual as well. The estimate is improving but has not converged:

r_1=\frac1{17\sqrt{17}}\begin{pmatrix}-12\\48\end{pmatrix},\qquad \|r_1\|_2=\frac{12}{17}

Practical Checks

These checks decide whether a computed eigenvalue and eigenvector pair is correct without repeating the whole determinant. Work through them in order for the pair you obtained.

Factor the shifted matrix with a stable method and pivoting where appropriate. If the shift lands exactly on an eigenvalue, the shifted matrix is singular and the solve must be handled or the shift adjusted.
Stop using a $residual norm$ . A small change in consecutive eigenvalue estimates alone does not prove that the eigenpair is accurate.
Eigenvector sign is arbitrary: $q$ and $-q$ represent the same eigendirection.
For clustered eigenvalues or general nonnormal matrices, use algorithms designed for multiple vectors or Schur form, using Lanczos for symmetric problems and Arnoldi or $\mathrm{QR}$ -based eigensolvers.