Python Inverse of a Matrix

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.

Solution 1:

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.

Solution 2:

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.

Solution 3:

It is a pity that the chosen matrix, repeated here again, is either singular or badly conditioned:

A = matrix( [[1,2,3],[11,12,13],[21,22,23]])

By definition, the inverse of A when multiplied by the matrix A itself must give a unit matrix. The A chosen in the much praised explanation does not do that. In fact just looking at the inverse gives a clue that the inversion did not work correctly. Look at the magnitude of the individual terms - they are very, very big compared with the terms of the original A matrix...

It is remarkable that the humans when picking an example of a matrix so often manage to pick a singular matrix!

I did have a problem with the solution, so looked into it further. On the ubuntu-kubuntu platform, the debian package numpy does not have the matrix and the linalg sub-packages, so in addition to import of numpy, scipy needs to be imported also.

If the diagonal terms of A are multiplied by a large enough factor, say 2, the matrix will most likely cease to be singular or near singular. So

A = matrix( [[2,2,3],[11,24,13],[21,22,46]])

becomes neither singular nor nearly singular and the example gives meaningful results... When dealing with floating numbers one must be watchful for the effects of inavoidable round off errors.