Find a matrix from its eigenvalues and corresponding vectors

Question

Suppose $A$ is a $3 \times 3$ matrix with eigenvalues $\lambda_1=-1$ $\lambda_2=0$ and $\lambda_3=1$ and with the corresponding eigenvectors $\vec{v_1}=<1,0,2>$ $\vec{v_2}=<-1,1,0>$ and $\vec{v_3}=<0,0,1>$

Find matrix the $A$

So I made $P=$ $\begin{bmatrix} 1 & -1 & 0 \\ 0 & 1 & 0 \\ 2 & 0 & 1 \\ \end{bmatrix}$ and got $P^{-1} =$ $\begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ -2 & -2 & 1 \\ \end{bmatrix}$

I am unsure where to go from here though? I feel as though maybe there is missing information in the question?

Answer 1

The searched matrix is $M=PDP^{-1}$ where $D$ is the diagonal matrix that has as diagonal elements the eigenvalues, in the same order as the eigenvectors in $P$ (see here).

$$ M= \begin{bmatrix} 1&-1&0\\ 0&1&0\\ 2&0&1 \end{bmatrix} \begin{bmatrix} -1&0&0\\ 0&0&0\\ 0&0&1 \end{bmatrix} \begin{bmatrix} 1&1&0\\ 0&1&1\\ -2&-2&1 \end{bmatrix} = \begin{bmatrix} -1&-1&0\\ 0&0&0\\ -4&-4&1 \end{bmatrix} $$

Answer 2

Hint:

$$\begin{pmatrix}a&b&c\\d&e&f\\g&h&i\end{pmatrix}\begin{pmatrix}1\\0\\2\end{pmatrix}=\begin{pmatrix}-1\\0\\-2\end{pmatrix}$$

and the same with other eigenvalues and eigenvectors.

Answer 3

$A=P \Lambda P^{-1}$, where $\Lambda=diag(\lambda_1,\dots,\lambda_3)$