Python Inverse of a Matrix

Question

How do I get the inverse of a matrix in python? I\'ve implemented it myself, but it\'s pure python, and I suspect there are faster modules out there to do it.

Answer 1

If you hate numpy, get out RPy and your local copy of R, and use it instead.

(I would also echo to make you you really need to invert the matrix. In R, for example, linalg.solve and the solve() function don't actually do a full inversion, since it is unnecessary.)

Answer 2

For those like me, who were looking for a pure Python solution without pandas or numpy involved, check out the following GitHub project: https://github.com/ThomIves/MatrixInverse.

It generously provides a very good explanation of how the process looks like "behind the scenes". The author has nicely described the step-by-step approach and presented some practical examples, all easy to follow.

This is just a little code snippet from there to illustrate the approach very briefly (AM is the source matrix, IM is the identity matrix of the same size):

def invert_matrix(AM, IM):
    for fd in range(len(AM)):
        fdScaler = 1.0 / AM[fd][fd]
        for j in range(len(AM)):
            AM[fd][j] *= fdScaler
            IM[fd][j] *= fdScaler
        for i in list(range(len(AM)))[0:fd] + list(range(len(AM)))[fd+1:]:
            crScaler = AM[i][fd]
            for j in range(len(AM)):
                AM[i][j] = AM[i][j] - crScaler * AM[fd][j]
                IM[i][j] = IM[i][j] - crScaler * IM[fd][j]
    return IM

But please do follow the entire thing, you'll learn a lot more than just copy-pasting this code! There's a Jupyter notebook as well, btw.

Hope that helps someone, I personally found it extremely useful for my very particular task (Absorbing Markov Chain) where I wasn't able to use any non-standard packages.

Answer 3

You should have a look at numpy if you do matrix manipulation. This is a module mainly written in C, which will be much faster than programming in pure python. Here is an example of how to invert a matrix, and do other matrix manipulation.

from numpy import matrix
from numpy import linalg
A = matrix( [[1,2,3],[11,12,13],[21,22,23]]) # Creates a matrix.
x = matrix( [[1],[2],[3]] )                  # Creates a matrix (like a column vector).
y = matrix( [[1,2,3]] )                      # Creates a matrix (like a row vector).
print A.T                                    # Transpose of A.
print A*x                                    # Matrix multiplication of A and x.
print A.I                                    # Inverse of A.
print linalg.solve(A, x)     # Solve the linear equation system.

You can also have a look at the array module, which is a much more efficient implementation of lists when you have to deal with only one data type.

Answer 4

Make sure you really need to invert the matrix. This is often unnecessary and can be numerically unstable. When most people ask how to invert a matrix, they really want to know how to solve Ax = b where A is a matrix and x and b are vectors. It's more efficient and more accurate to use code that solves the equation Ax = b for x directly than to calculate A inverse then multiply the inverse by B. Even if you need to solve Ax = b for many b values, it's not a good idea to invert A. If you have to solve the system for multiple b values, save the Cholesky factorization of A, but don't invert it.

See Don't invert that matrix.

Answer 5

Numpy will be suitable for most people, but you can also do matrices in Sympy

Try running these commands at http://live.sympy.org/

M = Matrix([[1, 3], [-2, 3]])
M
M**-1

For fun, try M**(1/2)

Answer 6

You could calculate the determinant of the matrix which is recursive and then form the adjoined matrix

Here is a short tutorial

I think this only works for square matrices

Another way of computing these involves gram-schmidt orthogonalization and then transposing the matrix, the transpose of an orthogonalized matrix is its inverse!