Finding the eigenvalues of a matrix can be done by solving the characteristic equation.
Here's the matrix I will be referring to.
Solving the characteristic equation
To find the eigenvalues of the above A, we solve the characteristic equation, which is defined as follows.
In this case, $I$ is simply the identity matrix. Swapping in $A$ and $I$ yields the following.
$1$ times $λ$ is just $λ$, so we can clarify further.
Then, we do the simple subtraction.
Remembering how to find the determinant, we can then express this as follows.
This can be rearranged to a tidier expression.
This is now the characteristic polynomial, with solutions at 5 and -1: these are the eigenvalues of $A$.
This is the method for returning eigenvalues in Numpy. Note that the similar
numpy.linalg.eig() method returns both the eigenvalues and the eigenvectors.