Eigendecomposition question: obvious probably, but having trouble seeing it

Question

$A$ is a $2×2$ matrix, $S$ is the $2×2$ matrix where the columns are the two eigenvectors of $A$ and $M$ is the $2×2$ matrix with the two eigenvalues along the diagonal and zeros elsewhere. I am reading that by the definition of eigenvalues and eigenvectors, we have the following identity: $$AS=SM$$ Why is that true? Thank you.

Answer 1

Let $v_1,v_2 $ be your eigenvectors. We can write down the matrix $S$ with $v_1,v_2 $ as columns.

If you write $S=\begin{bmatrix} v_1 & v_2 \end{bmatrix}$ then $AS=\begin{bmatrix} Av_1 & Av_2 \end{bmatrix}$,

on the other side $M=\begin{bmatrix} m_1 & 0 \\ 0 & m_2 \end{bmatrix}$ then $\begin{bmatrix} v_1 & v_2 \end{bmatrix} \begin{bmatrix} m_1 & 0 \\ 0 & m_2 \end{bmatrix}= \begin{bmatrix} m_1 v_1 & m_2v_2 \end{bmatrix}$.

So the equation means $\begin{bmatrix} Av_1 & Av_2 \end{bmatrix}= \begin{bmatrix} m_1 v_1 & m_2v_2 \end{bmatrix}$.

Answer 2

The diagonalisation of $A$ is $SMS^{-1}$. Right-multiplying by $S$ gives $AS=SM$.

(This only works if $A$ is diagonalisable in the first place.)