Let A be a square matrix. Let’s say A has k eigenvalues λ1, λ2, … λk. And the corresponding eigenvectors are X1, X2, … Xk.

Now, let’s define the matrix P as:

In other words, the columns of the matrix P are the eigenvectors X1, X2, … Xk. Let’s also define a diagonal matrix D as:

At this point, we know that if we multiply the matrix by an eigenvector of the matrix, we get the eigenvalue times the eigenvector.

So, if we multiply A with the matrix P, we get the following:

So,

In other words, if A is a square matrix with k linearly independent eigenvectors, then A can be factorized like the following:

where P is a matrix in which the columns are the eigenvectors of A, and D is a diagonal matrix with the eigenvalues of A in its principal diagonal.

In fact, we can also prove by induction that:

Please note that D is a diagonal matrix. So, it is mathematically convenient to calculate Dⁿ.

So, if A can be eigen decomposed and none of its eigenvalues are zero, then A is invertible. And the inverse of A can be calculated as:

Eigen decomposition of a square matrix using Python

We can use the following Python code to calculate the eigen decomposition of a square matrix A…