Automatically populating matrix elements in SymPy

Question

Is there a way to implicitly define the elements of a symbolic matrix in SymPy following a rule such as: symbol followed by subindices in the matrix (or pairs of numbers)

For example, I would like to define a 3 x 2 matrix called M, and I would like SymPy to automatically create it and populate it as:

M = 
[ M_11 M_12]
[ M_21 M_22]
[ M_31 M_32]

If there is no way to do this implicitly, what would be the easiest way to do this explicitly (e.g. looping)?

Answer 1

Consider using the MatrixSymbol rather than Matrix object. MatrixSymbol represents matrices without the need for explicit elements.

In [1]: M = MatrixSymbol('M', 3, 2)

In [2]: M  # Just an expression
Out[2]: M

In [3]: Matrix(M)  # Turn it into an explicit matrix if you desire
Out[3]: 
⎡M₀₀  M₀₁⎤
⎢        ⎥
⎢M₁₀  M₁₁⎥
⎢        ⎥
⎣M₂₀  M₂₁⎦


In [4]: M.T * M   # Still just an expression
Out[4]: 
 T  
M ⋅M

In [5]: Matrix(M.T * M)  # Fully evaluate
Out[5]: 
⎡       2      2      2                                  ⎤
⎢    M₀₀  + M₁₀  + M₂₀        M₀₀⋅M₀₁ + M₁₀⋅M₁₁ + M₂₀⋅M₂₁⎥
⎢                                                        ⎥
⎢                                    2      2      2     ⎥
⎣M₀₁⋅M₀₀ + M₁₁⋅M₁₀ + M₂₁⋅M₂₀      M₀₁  + M₁₁  + M₂₁      ⎦

Answer 2

How about something like this:

import sympy

M = sympy.Matrix(3, 2, lambda i,j:sympy.var('M_%d%d' % (i+1,j+1)))

Edit: I suppose I should add a small explanation. The first two arguments to sympy.Matrix() are defining the matrix as 3x2 (as you specified). The third argument is a lambda function, which is essentially a shorthand way of defining a function in one line, rather than formally defining it with def. This function takes variables i and j as input, which conveniently are the indices of the matrix. For each pair (i,j) which are passed into the lambda (i.e., for each element of the matrix), we are creating a new symbolic variable M_ij. sympy.var() takes a string as input which defines the name of the new symbolic variable. We generate this string on-the-fly using the format string 'M_%d%d' and filling it with (i+1,j+1). We are adding 1 to i and j because you want the matrix to be 1-indexed, rather than 0-indexed as is the standard in Python.