# 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?

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.