# Doubts on inverse power method

I found written that if matrix A is real and you use the Power method to find eigenvalues then "If the matrix and starting vector are real then the power method can never give a result with an imaginary part." reference.

Is it also true for the inverse power method used to find a better approximation of the eigenvalue given a initial approximation? I've written a simple MATLAB program and I think it's false but I need some clarification.

What about the initial approximation of complex eigenvalue? Should it be complex in order to converge?

## 1 Answer

Inverse iteration will also stay real along the way. Finding complex eigenvalues is tricky; either your method needs to make a block matrix like $\begin{bmatrix} a & -b \\ b & a \end{bmatrix}$ show up by itself, or else it needs to give complex eigenvalues "a shot", by looking for the eigenvalues of $A-\lambda I$ for some complex number $\lambda$ or by looking at complex starting vectors.

• I'm not sure to have correctly understood... What's the difference if the initial vector is real or complex? – sunrise Feb 26 '16 at 20:41
• @sunrise If the initial vector is complex and "random" then it will have a component in the direction of the smallest eigenvalue, which inverse iteration will amplify, so you will get the convergence (provided there is a strictly smallest eigenvalue). – Ian Mar 7 '16 at 14:07