Eigenvalues and Eigenvector of matrix

Question

I try to find the eigenvectors and eigenvalues from the matrix:

$$M =\pmatrix{1/5 & 2/5 \\ 2/5 & 4 /5}$$

I started like this:

$$M = \pmatrix{1/5 - \Delta & 2/5 \\ 2/5 & 4 /5 -\Delta}$$

$$|M| = (1/5 - \Delta) * (4 /5 -\Delta) - 2/5 * 2/5 $$ $$|M| = \Delta²- \Delta$$

But as you can see, i only can find $\Delta = 1$ as one of the eigenvalues! How should i build a eigenvector? Thanks

Answer 1

As you can see from the comments above the eigenvalues are $\Delta_1=0$ and $\Delta_2=1$.

By definition a vector $v\in \mathbb R^2$ is called an eigenvector of M belonging to the eigenvalue $\Delta$ if

$$Mv=\Delta v \iff Mv-\Delta v=0 \iff (M-\Delta I_2)v=0 $$

So the eigenvalues of M are the set of vectors in $\{v \in \mathbb R^2 :(M-\Delta_{1,2} I_2)v=0\}=ker(M-\Delta_{1,2} I_2)$

Note: The zero vector fullfils always the above condition. But in the most literature they require that $v \neq 0$

Answer 2

Hint: $$\Delta(\Delta -1) = 0 \longrightarrow \Delta = 1 \text{ or } \Delta = \text{ what? }$$