Eigenvectors of a matrix with only one value?

Question

So I'm trying to find the eigenvectors of $$ \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} $$

And the answer states that they are supposed to be: $$ \vec{v}_1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix} $$

How might I get the answer I need?

Answer 1

The characteristic polynomial is $(1-\lambda)(-\lambda) = 0$, so the possible eigenvalues are $\lambda = 0,1$. Recall that an eigenvector would have the property $Av = \lambda v$. So we are looking for vectors such that $Av=v$ or $Av = 0$.

The problem becomes to find the null space of the matrices $(A-I)$ and $A$ to get the space of eigenvectors corresponding to the eigenvalues $1$ and $0$ respectively.

Answer 2

When a matrix is already in diagonal form, as is your matrix, the eigenvectors and eigenvalues can be written down.

The vector $\begin{pmatrix}1\\0\\\end{pmatrix}$ has eigenvalue $1$ and the vector $\begin{pmatrix}0\\1\\\end{pmatrix}$ has eigenvalue $0$.

Note that $1$ and $0$ are the numbers on the diagonal and that $\begin{pmatrix}1\\0\\\end{pmatrix}$ and$\begin{pmatrix}0\\1\\\end{pmatrix}$ are the base vectors.