Introduction to eigenvalues and eigenvectors

If you watched the video from 3Blue1Brown, you should know the meaning of eigenvalues and eigenvectors by now.

In equation, it’s written like this:

In a matrix form, it looks something like this:

Just as a recap, eigenvectors are the vectors that does not change its orientation, but just scales by a factor of its corresponding eigenvalue.

To solve for eigenvalues and eigenvectors, here are the steps you need to take.

Let’s take a quick example using 2 x 2 matrix.

By solving the determinant = 0, we get the eigenvalues. Now we just need to consider each eigenvalue case separately.

Now you got one of the eigenvectors. Moving on to the next.

Great! Now you solved the eigenvalue and eigenvector problem!

Let’ take a look at the results and make sure that what 3Blue1Brown video was saying makes sense.

As you can see above, no matter what kind of transition matrix “A” you have, if you managed to find its eigenvalues and eigenvectors, the transition using the matrix “A