Matrix inversion without Numpy

For a 4 x 4 matrix it’s probably just about OK to use the mathematical formula, which you can find using Googling “formula for 4 by 4 matrix inverse”. For example here (I can’t vouch for its accuracy):

http://www.cg.info.hiroshima-cu.ac.jp/~miyazaki/knowledge/teche23.html

In general inverting a general matrix is not for the faint-hearted. You have to be aware of all the mathematically difficult cases and know why they won’t apply to your usage, and catch them when you are supplied with mathematically pathological inputs (that, or return results of low accuracy or numerical garbage in the knowledge that it won’t matter in your usage case provided you don’t actually end up dividing by zero or overflowing MAXFLOAT … which you might catch with an exception handler and present as “Error: matrix is singular or very close thereto”).

It’s generally better as a programmer to use library code written by numerical mathematics experts, unless you are willing to spend time understanding the physical and mathematical nature of the particular problem that you are addressing and become your own mathematics expert in your own specialist field.

Here is a more elegant and scalable solution, imo. It’ll work for any nxn matrix and you may find use for the other methods. Note that getMatrixInverse(m) takes in an array of arrays as input. Please feel free to ask any questions.

def transposeMatrix(m):
    return map(list,zip(*m))

def getMatrixMinor(m,i,j):
    return [row[:j] + row[j+1:] for row in (m[:i]+m[i+1:])]

def getMatrixDeternminant(m):
    #base case for 2x2 matrix
    if len(m) == 2:
        return m[0][0]*m[1][1]-m[0][1]*m[1][0]

    determinant = 0
    for c in range(len(m)):
        determinant += ((-1)**c)*m[0][c]*getMatrixDeternminant(getMatrixMinor(m,0,c))
    return determinant

def getMatrixInverse(m):
    determinant = getMatrixDeternminant(m)
    #special case for 2x2 matrix:
    if len(m) == 2:
        return [[m[1][1]/determinant, -1*m[0][1]/determinant],
                [-1*m[1][0]/determinant, m[0][0]/determinant]]

    #find matrix of cofactors
    cofactors = []
    for r in range(len(m)):
        cofactorRow = []
        for c in range(len(m)):
            minor = getMatrixMinor(m,r,c)
            cofactorRow.append(((-1)**(r+c)) * getMatrixDeternminant(minor))
        cofactors.append(cofactorRow)
    cofactors = transposeMatrix(cofactors)
    for r in range(len(cofactors)):
        for c in range(len(cofactors)):
            cofactors[r][c] = cofactors[r][c]/determinant
    return cofactors

As of at least July 16, 2018 Numba has a fast matrix inverse. (You can see how they overload the standard NumPy inverse and other operations here.)

Here are the results of my benchmarking:

import numpy as np
from scipy import linalg as sla
from scipy import linalg as nla
import numba

def gen_ex(d0):
  x = np.random.randn(d0,d0)
  return x.T + x

@numba.jit
def inv_nla_jit(A):
  return np.linalg.inv(A)

@numba.jit
def inv_sla_jit(A):
  return sla.inv(A)

For small matrices it is particularly fast:

ex1 = gen_ex(4)
%timeit inv_nla_jit(ex1) # NumPy + Numba
%timeit inv_sla_jit(ex1) # SciPy + Numba
%timeit nla.inv(ex1)     # NumPy
%timeit sla.inv(ex1)     # SciPy

[Out]

2.54 µs ± 467 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)
67.3 µs ± 9.18 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)
63.5 µs ± 7.65 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)
56.6 µs ± 5.03 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)

Notice that the speedup only works for NumPy inverse, not SciPy (as expected).

Slightly larger matrix:

ex2 = gen_ex(40)
%timeit inv_nla_jit(ex2) # NumPy + Numba
%timeit inv_sla_jit(ex2) # SciPy + Numba
%timeit nla.inv(ex2)     # NumPy
%timeit sla.inv(ex2)     # SciPy

[Out]

131 µs ± 12.9 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)
278 µs ± 26.2 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)
231 µs ± 24.5 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)
189 µs ± 11.2 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)

So there’s still a speedup here but SciPy is catching up.

Here is another way, using gaussian elimination instead:

def eliminate(r1, r2, col, target=0):
    fac = (r2[col]-target) / r1[col]
    for i in range(len(r2)):
        r2[i] -= fac * r1[i]

def gauss(a):
    for i in range(len(a)):
        if a[i][i] == 0:
            for j in range(i+1, len(a)):
                if a[i][j] != 0:
                    a[i], a[j] = a[j], a[i]
                    break
            else:
                raise ValueError("Matrix is not invertible")
        for j in range(i+1, len(a)):
            eliminate(a[i], a[j], i)
    for i in range(len(a)-1, -1, -1):
        for j in range(i-1, -1, -1):
            eliminate(a[i], a[j], i)
    for i in range(len(a)):
        eliminate(a[i], a[i], i, target=1)
    return a

def inverse(a):
    tmp = [[] for _ in a]
    for i,row in enumerate(a):
        assert len(row) == len(a)
        tmp[i].extend(row + [0]*i + [1] + [0]*(len(a)-i-1))
    gauss(tmp)
    ret = []
    for i in range(len(tmp)):
        ret.append(tmp[i][len(tmp[i])//2:])
    return ret

Inverse matrix of 3×3 without numpy [python3]

import pprint


def inverse_3X3_matrix():
    I_Q_list = [[0, 1, 1],
                [2, 3, -1],
                [-1, 2, 1]]
    det_ = I_Q_list[0][0] * (
            (I_Q_list[1][1] * I_Q_list[2][2]) - (I_Q_list[1][2] * I_Q_list[2][1])) - 
           I_Q_list[0][1] * (
                   (I_Q_list[1][0] * I_Q_list[2][2]) - (I_Q_list[1][2] * I_Q_list[2][0])) + 
           I_Q_list[0][2] * (
                   (I_Q_list[1][0] * I_Q_list[2][1]) - (I_Q_list[1][1] * I_Q_list[2][0]))
    co_fctr_1 = [(I_Q_list[1][1] * I_Q_list[2][2]) - (I_Q_list[1][2] * I_Q_list[2][1]),
                 -((I_Q_list[1][0] * I_Q_list[2][2]) - (I_Q_list[1][2] * I_Q_list[2][0])),
                 (I_Q_list[1][0] * I_Q_list[2][1]) - (I_Q_list[1][1] * I_Q_list[2][0])]

    co_fctr_2 = [-((I_Q_list[0][1] * I_Q_list[2][2]) - (I_Q_list[0][2] * I_Q_list[2][1])),
                 (I_Q_list[0][0] * I_Q_list[2][2]) - (I_Q_list[0][2] * I_Q_list[2][0]),
                 -((I_Q_list[0][0] * I_Q_list[2][1]) - (I_Q_list[0][1] * I_Q_list[2][0]))]

    co_fctr_3 = [(I_Q_list[0][1] * I_Q_list[1][2]) - (I_Q_list[0][2] * I_Q_list[1][1]),
                 -((I_Q_list[0][0] * I_Q_list[1][2]) - (I_Q_list[0][2] * I_Q_list[1][0])),
                 (I_Q_list[0][0] * I_Q_list[1][1]) - (I_Q_list[0][1] * I_Q_list[1][0])]

    inv_list = [[1 / det_ * (co_fctr_1[0]), 1 / det_ * (co_fctr_2[0]), 1 / det_ * (co_fctr_3[0])],
                [1 / det_ * (co_fctr_1[1]), 1 / det_ * (co_fctr_2[1]), 1 / det_ * (co_fctr_3[1])],
                [1 / det_ * (co_fctr_1[2]), 1 / det_ * (co_fctr_2[2]), 1 / det_ * (co_fctr_3[2])]]

    pprint.pprint(inv_list)


inverse_3X3_matrix()

Simply add all methods

import math

def getMinorIndex(matrixLocal, x, y):
  minor = []
  for i in range(3):
    minorRow = []
    if i == x:
      continue
    for j in range(3):
      if j == y:
        continue
      minorRow.append(matrixLocal[i][j])
    minor.append(minorRow)
  return minor

def getDeterminant2By2(matrixLocal):
  determinant = matrixLocal[0][0] * matrixLocal[1][1] - matrixLocal[0][1] * matrixLocal[1][0]
  return determinant

def getDeterminant(matrixLocal):
  determinant = 0
  for x in range(3):
    t = getDeterminant2By2(getMinorIndex(matrixLocal, 0, x))
    e = matrixLocal[0][x]
    determinant += (t * e * math.pow(-1, x))
  return determinant

def getCofactorMatrix(matrixLocal):
  cofactorMatrix = []
  for i in range(3):
    row = []
    for j in range(3):
      e = matrixLocal[i][j]
      t = getDeterminant2By2(getMinorIndex(matrixLocal, i, j))
      row.append(t * math.pow(-1, i + j))
    cofactorMatrix.append(row)
  return cofactorMatrix

def transpose(matrixLocal):
  transposeMatrix = []
  for i in range(3):
    row = []
    for j in range(3):
      e = matrixLocal[j][i]
      row.append(e)
    transposeMatrix.append(row)
  return transposeMatrix

def divideMatrix(matrixLocal, divisor):
  ansMatrix = []
  for i in range(3):
    row = []
    for j in range(3):
      e = matrixLocal[i][j]/divisor
      row.append(e)
    ansMatrix.append(row)
  return ansMatrix

cofactor = getCofactorMatrix(matrix)
adjoint = transpose(cofactor)
det = getDeterminant(matrix)
inverse = divideMatrix(adjoint, det)
inverse

I found that Gaussian Jordan Elimination Algorithm helped a lot when attempting this. If you’re going to use a given matrix (any size, i.e 5×5) where the hardcore formula for it is 49 pages long. It’s best to use this. To inverse a matrix place it as a 2D array and then run the Inverse function

# Python test Guassion Jordan Elimination
# Inputs are 2D array not matrix

Test_Array = [[3,3,2,1,1],[2,1,3,2,3],[1,3,3,2,2],[2,3,3,1,1], 
[3,1,2,1,2]]

# Creating storage & initalizing for augmented matrix
# this is the same as the np.zeros((n,2*n)) function
def nx2n(n_Rows, n_Columns):
    Zeros = []
    for i in range(n_Rows):
        Zeros.append([])
        for j in range(n_Columns*2):
            Zeros[i].append(0)
    return Zeros

# Applying matrix coefficients
def update(inputs, n_Rows, n_Columns, Zero):
    for i in range(n_Rows):
        for j in range(n_Columns):
            Zero[i][j] = inputs[i][j]
    return Zero

# Augmenting Identity Matrix of Order n
def identity(n_Rows, n_Columns, Matrix):
    for i in range(n_Rows):
        for j in range(n_Columns):
            if i == j:
                Matrix[i][j+n_Columns] = 1
    return Matrix

# Applying & implementing the GJE algorithm
def Gussain_Jordan_Elimination(n_Rows, n_Columns, Matrix):
    for i in range(n_Rows):
        if Matrix[i][i] == 0:
            print('error cannot divide by "0"')
    
        for j in range(n_Columns):
            if i != j:
                ratio = Matrix[j][i]/Matrix[i][i]

                for k in range(2*n_Columns):
                    Matrix[j][k] = Matrix[j][k] - ratio * Matrix[i][k]
    return Matrix

# Row Operation to make Principal Diagonal Element to '1'
def row_op(n_Rows, n_Columns, Matrix):
    for i in range(n_Rows):
        divide = Matrix[i][i]
        for j in range(2*n_Columns):
            Matrix[i][j] = Matrix[i][j]/divide
    return Matrix

# Display Inversed Matix
def Inverse(Matrix):
    returnable = []
    number_Rows = int(len(Matrix))
    number_Columns = int(len(Matrix[0]))
    Inversed_Matrix = (row_op(number_Rows, number_Columns, 
        Gussain_Jordan_Elimination(number_Rows, number_Columns, 
        identity(number_Rows, number_Columns, 
        update(Matrix, number_Rows, number_Columns, 
        nx2n(number_Rows, number_Columns))))))

    for i in range(number_Rows):
        returnable.append([])
        for j in range(number_Columns, 2*number_Columns):
            returnable[i].append(Inversed_Matrix[i][j])
    return returnable

print(Inverse(Test_Array))