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

User · Answer

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  quot behind the scenes quot   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

User · Answer

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

User · Answer

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

User · Answer

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

User · Answer

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

User · Answer

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   Thanks for your contribution   OldAl

User · Answer

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

User · Answer

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

User · Answer

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

User · Answer

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

User · Answer

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  quot behind the scenes quot   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

User · Answer

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

User · Answer

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

User · Answer

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   Thanks for your contribution   OldAl

User · Answer

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

User · Answer

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

User · Answer

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

User · Answer

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

User · Answer

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

User · Answer

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

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