Inverse Power Method

We know that if $\lambda$ is an eigenvalue of $A$ and if $A$ is nonsingular then $\lambda^{-1}$ is an eigenvalue of $A^{-1}$.

This suggests a way to estimate the smallest eigenvalue of $A$ using the power method: arrange eigenvalues as

$$\vert\lambda_1\vert\ge \vert\lambda_2\vert\ge\ldots \le \vert\lambda_{n-1}\vert>\vert\lambda_n\vert>0$$

can be done since $A$ is non-singular, $0$ is not an eigenvalue. The eigenvalues of $A^{-1}$ arranged:

$$\vert\lambda^{-1}_n\vert>\vert\lambda^{-1}_{n-1}\vert\ge \cdots \ge \vert\lambda_1 ^{-1_1}>0$$

$\therefore$ Apply power method to $A^{-1}$. But we don't compute $A^{-1}$. Instead solve

$$Ax^{(k+1)}=x^{(k)}$$

for $x^{(k+1)}$ by some efficient linear algebra solver. One could consider an $LU$ factorization, since it only has to be done only once.

These two suggest a way to find the eigenvalue farthest to a given value $\mu$. The ``Shifted Matrix Power Method,'' here $\mu$ is complex generally: the trick is to construct a matrix

$$\hat A=(A-\mu I)$$

and then use the regular power method on $\hat {A}$, i.e.

$$x^{(k+1)}=\hat Ax^{(k)}$$

Finally, we could consider the eigenvalue closest to $\mu$. In this case we apply the inverse power method on $\hat A$, i.e.

$$\hat A x^{(k+1)}=x^{(k)}$$

$\Box$