In every example so far, the components of each vector along the eigen-axes have been computed for you. This was done to allow you to focus on how they work, rather than you habitually stopping your thoughts so you can calulate them. But of course, one must learn how to calculate them. Given the coordinates $(x, y)$ for our column vector using the regular axes, we need to find new coordinates $x', y'$ with the property that

This can be solved very quickly with a special matrix $P$ with columns $\vec{v}_1$ and $\vec{v}_2$:

And there you go! Take your matrix of eigenvectors, find its inverse, and apply it to your standard coordinates.

Once you know how to change from regular coordinates to eigen-coordinates, you can diagonalize your matrix. Digaonal matrices are the matrices that scale each of the regular axes. The eigen-value for each axis is on that diagonal. For example, if the $x$-axis is scaled by $2$ and the $y$-axis is scaled by 3, then we can quickly see what the matrix has to be:

Now for other eigen-axes, you just use the matrix of eigenvectors

to switch between your axes like this: If $A$ is a given matrix and $D = \begin{bmatrix} \lambda_1 & 0 \\ 0 & \lambda_2 \end{bmatrix}$ is its diagonal form, then

That's it, you've diagonalized $A$!