We calculate the eigenvalues and eigenvectors of the matrix
(1) 

numpy.linalg.eig
function returns a tuple consisting
of a vector and an array. The vector (here w
) contains
the eigenvalues. The array (here v
) contains the
corresponding eigenvectors, one eigenvector per column. The eigenvectors
are normalized so their Euclidean norms are 1.
The eigenvalue w[0]
goes with the 0
th column
of v
. The eigenvalue w[1]
goes with column
1
, etc. To extract the ith column vector, we use

Just to be clear about what is happening here, let's check the
eigenvector/eigenvalue condition for the second eigenvalue and
eigenvector.
(2) 
v[:,1]
by A
and check
that it is the same as multiplying the same eigenvector by its
eigenvalue w[1]
.
