## Diagonalization

1. The problem statement, all variables and given/known data

I think my teacher made a mistake in his homework answer. I need to verify this for practice. The answer I got is below. The answer the teacher has is in the pdf.

2. Relevant equations

Please refer to attached pdf

3. The attempt at a solution

So there is two eigenvalues= 4 and 2
but the eigenvalue 2 has 2 eigenvectors [-1 1 0]T and [0 0 1]T but my teacher has only one [-1 1 0]T. That's why he says A is not diagonalizable. Do you think it's correct?
Attached Thumbnails

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 I get the same result as your teacher for $$\lambda = 2$$ $$A= \begin{bmatrix} 3 & 1 & 0\\ 0 & 2 & 1\\ 1 & 1 & 3 \end{bmatrix}$$ so for lamda = 2, $$(A-2I)\vec{v}=\vec{0}$$ $$\begin{bmatrix} 1 & 1 & 0\\ 0 & 0 & 1\\ 1 & 1 & 1 \end{bmatrix} \begin{bmatrix} v_1\\ v_2\\ v_3 \end{bmatrix} = \begin{bmatrix} 0\\ 0\\ 0\\ \end{bmatrix}$$ I've personally always found it easier to not do row operations here and just jump straight in. From that you get $$v_3 = 0$$ and $$v_1 = -v_2$$ so therefore $$\vec{v} = \begin{bmatrix} 1\\ -1\\ 0 \end{bmatrix}$$ Since it is a 3x3 matrix, it needs 3 eigenvectors to be diagonalizable. Hope this helps.
 Hi, thanks for replying. Attached is how I got my vectors, do you think my steps are correct. Attached Thumbnails

Mentor

## Diagonalization

No. Why do you think (0, 0, 1) is an eigenvector? You seem to just pull that out of thin air.

 You seem to have made a mistake in the step $$(A-2I)\vec{v}=\vec{0}$$ Why is your second row 1 0 1 instead of 0 0 1 ? Have you said $$v_2= \delta$$ ? I'm not sure if I have read that correctly. If it is a delta, you can't say that unless the whole row equals zero. i.e. $$\begin{bmatrix} 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} v_1 \end{bmatrix} = \begin{bmatrix} 0 \end{bmatrix}$$

 Tags algebra, diagonalization, eigenvalue, linear
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