if we know eigenvector of a matrix, what will be eigenvector for cube of matrix?

Question

How can I verify $(1,- {1 \over 2},0)$ is an eigenvector of M³?

I can consider a diagonal matrix with all diagonal entries $⁻3$, or I can consider a general diagonal matrix.

But my question is how I can shortly and quickly have an idea about eigenvector for M³?

Answer 1

$Mv=3v\implies M(-\frac12v)=-\frac32v\implies M^3(-\frac12v)=-\frac{27}2v$. So we get $-\frac{27}2\begin{pmatrix}-2\\1\\0\end{pmatrix}=\begin{pmatrix}27\\-\frac{27}2\\0\end{pmatrix}$.

Answer 2

Think of the matrix $M$ as a linear operator. What happens when you apply $M$ three times to an eigenvector? Do it one step at a time and I think the answer will jump out at you.