review on linear algebra

Think of a linear transformation, it rotate and shear your original coordinate
$
begin{bmatrix}
3 & 1
0 & 2
end{bmatrix} tag{0}
$

Now we look at a particular case: when it make a effect on a vector(from her coordinate system to our …), and the vector remain in the same span

$P^{-1}MP = Sigma$
where p is M’s eigenvectors in column, meaning a vector $vec{v}$ will first transform from P’s coordinate system into ours and then go into linear transformation M and return to P’s coordinate system.
The special thing about eigenvector is that it remain in the same direction and only scale its size. So the P’s base vector only scales, and the result is that the total transformation will be diagonal matrix
$
begin{bmatrix}
lambda{1} & 0
0 & lambda{2}
end{bmatrix} tag{1}
$
(diagonal matrix means that it only scales the base vectors)