Matrix diagonalization. Can't see what I am doing wrong

Question

I have to diagonolize a matrix A \begin{bmatrix}0&-3&-1&1\\2&5&1&-1\\-2&-3&1&1\\2&3&1&1\end{bmatrix}

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I do $det(A-λ)=0$ and I get $λ_{1}=1$, $λ_{2}=2$, $λ_{3}=2$, $λ_{4}=2$

So possible diagonal matrix looks like: \begin{bmatrix}1&0&0&0\\0&2&0&0\\0&0&2&0\\0&0&0&2\end{bmatrix}

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Then I look for eigenvectors for $λ=1$ and $λ=2$

I get:

For $λ=1$: $(1, -1, 1,-1)$

For $λ=2$: $(1, 0, 0, 2)$, $(0, 1, 0, 3)$, $(0, 0, 1, 1)$

I create matrix P made of eigenvectors (as columns):

\begin{bmatrix}1&1&0&0\\-1&0&1&0\\1&0&0&1\\-1&2&3&1\end{bmatrix}

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And when checking if the diagonal matrix is correct by using formula:

$A=P^{-1}*D*P$

I don't get the correct answer :(

Can someone check my steps? I tried to do that a few times but still I am not getting the correct result.

Answer 1

Your computations are totally correct, but you made a tiny mistake in the order of the decomposition of $A$. It has to be $A = PDP^{-1}$.

Just remind yourself, what if A was e.g. symmetric, since then one could find orthogonal transformations and there you immediately see, that the most right matrix has to contain the eigenvectors in a horizontal manner and the left matrix has to contain them standing.

Answer 2

It seems you have not made any errors when computing the eigenvectors. I checked your results on the following matrix calculator: is $P$ the correct transformation matrix?

Also as was pointed out, you must use the correct formula for going from one basis to the other. The matrix $P$ is a transformation from the eigenvectors basis to the canonical one, not the other way around, i.e. the formula is $A = P D P^{-1}$.