How to get An eigenvalue and eigenvectors of a matrix that contain both zero column and zero row?

Question

Could anyone help in how to get the eigenvalue and eigenvectors of a matrix that contain both zero column and zero row like : \begin{pmatrix} -1 & 1 & 0\\ 1 & -1 & 0\\ 0 & 0 & 0\\ \end{pmatrix} Thnx In Advance.

Answer 1

We find the eigenvalues and eigenvectors as usual.

We have eigenvalues $|A - \lambda I| = 0$ and arrive at:

$$\lambda_{1,2,3} = -2, 0, 0$$

We have eigenvectors from $[A - \lambda_i I]v_i = 0$ and get:

$$v_1 = (-1,1,0), v_2 = (0,0,1), v_3 = (1,1,0)$$

We were lucky and did not need to find generalized eigenvectors.

Answer 2

It takes only a very mild form f curiosity to try out what happens if you multiply this matrix with a vector that is nonzero only in its last component.

Answer 3

Take the characteristic polynomial as usual and we get:

$-\lambda((1+\lambda)^2-1)=0\Rightarrow \lambda(\lambda^2+2\lambda)=0\Rightarrow \lambda=0,0,-2$ are the three eigenvalues.

for $\lambda=0$, we get eigen vectors satisfying: $x=y$, $z=z$, so eigenvectors are of the form $(1,1,\alpha)^T$ for $\alpha\in\mathbb{R}$.

For $\lambda=-2$ we get eigen vectors satisfying: $x=-y$, $z=z$ , so eigenvectors are of the form $(1,-1,\alpha)^T$ for $\alpha\in\mathbb{R}$