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Prop: If $T : V \rightarrow V$ and if $v_1,...,v_n$ is a basis such that $T$ is diagonal, then the $v_i$ are eigenvectors.

Pf: The eigenvalues of a diagonal matrix are its diagonal entries.

Let $T = \begin{pmatrix} a_1 & 0 & \dots & 0 \\ 0 & a_2 & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots &a_n \end{pmatrix}$, where the columns of $[T]$ are basis vectors.

$Tv_i = \begin{pmatrix} a_1 & 0 & \dots & 0 \\ 0 & a_2 & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots &a_n \end{pmatrix}\begin{pmatrix} a_1 \\ 0 \\ \vdots \\ 0\end{pmatrix}=\begin{pmatrix} a^2_1 \\ 0 \\ \vdots \\ 0\end{pmatrix}=\lambda\begin{pmatrix} a_1 \\ 0 \\ \vdots \\ 0 \end{pmatrix} =\lambda v_i \square$

Does my proof look right?

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    $\begingroup$ What exactly is your question? $\endgroup$ – Severin Schraven Dec 30 '19 at 18:33
  • $\begingroup$ Then you should formulate this in your post and also put the proof-verification tag :) $\endgroup$ – Severin Schraven Dec 30 '19 at 18:35
  • $\begingroup$ Your second equality is false. $\endgroup$ – amd Dec 30 '19 at 19:04
  • $\begingroup$ It is interesting to note that this means that, up to the order the eigenvalues occur in, there is only one diagonal matrix that a matrix can be similar to. That is, the diagonalization, if it exists, is essentially unique. See the Jordan-Chevalley normal form. This is a more general form. $\endgroup$ – Chris Custer Dec 30 '19 at 20:47
  • $\begingroup$ Sorry, it's Jordan-Chevalley decomposition. $\endgroup$ – Chris Custer Dec 30 '19 at 21:03
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I can't understand your proof.

Anyhow, we want to show that $Tv_i= \lambda_i v_i$ for some $\lambda_i$ a scalar, for every $i$.

Now $Tv_i= \begin{pmatrix} a_1 & 0 & \dots & 0 \\ 0 & a_2 & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots &a_n \end{pmatrix} \begin{pmatrix} 0 \\ \vdots \\ 1 \\ \vdots \\ 0 \end{pmatrix}=\begin{pmatrix} 0 \\ \vdots \\ a_i \\ \vdots \\ 0 \end{pmatrix}=a_i\begin{pmatrix} 0 \\ \vdots \\ 1 \\ \vdots \\ 0 \end{pmatrix}=a_iv_i$

So every $v_i$ is an eigenvector with eigenvalue $a_i$

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