# How to define a matrix with variables in SageMath?

I would like to define a function which produce matrices with variables. It will be something like the following.

def MatrixG(x,n,m):
r=Matrix(n,m)
for i in [0..n-1]:
for j in [0..m-1]:
r[i,j]=x[i,j]
return r


Here x[i,j]'s are variables. How to write the above correctly in Sage?

I also would like to have LU decomposition of some matrix like r1=MatrixG(3,3), r1 is a symbolic matrix.

Thank you very much.

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Sort by » oldest newest most voted Symbolic matrix:

sage: matrix(SR, 3, 2, lambda i,j: var('x_{}_{}'.format(i,j)))
[x_0_0 x_0_1]
[x_1_0 x_1_1]
[x_2_0 x_2_1]


SageMath cannot compute a LU decomposition since the symbolic ring is not exact.

Matrix with entries in a fraction field:

sage: R = PolynomialRing(QQ, 3, 2, var_array='x').fraction_field()
sage: A = matrix(R, 3, 2, R.gens()); A
[x00 x01]
[x10 x11]
[x20 x21]


LU decomposition:

sage: P, L, U = A.LU()
sage: P
[1 0 0]
[0 1 0]
[0 0 1]
sage: L
[                                        1                                         0                                         0]
[                                  x10/x00                                         1                                         0]
[                                  x20/x00 (-x01*x20 + x00*x21)/(-x01*x10 + x00*x11)                                         1]
sage: U
[                     x00                      x01]
[                       0 (-x01*x10 + x00*x11)/x00]
[                       0                        0]
sage: A == P*L*U
True


In the fraction field every nonzero element is invertible, and the LU decomposition algorithm did use some inverses, so the resulting "general formula" in fact has limited applicability when the variables are replaced by numbers (limited to those situations where the inverses exist). For example the inverse of x00 is used, so the formulas are not valid when x00 = 0. But you can repeat the calculation for that case:

sage: B = copy(A)
sage: B[0,0] = 0
sage: B.LU()
...


etc.

more

@rburing, thank you very much. I tried to use your codes: R = PolynomialRing(QQ, 3, 2, var_array='x').fraction_field() A = matrix(R, 3, 2, R.gens()); A

But it is said that "__init__() got an unexpected keyword argument 'var_array'". I used SageMath 9.0. Do you know how to fix this?

@rburing, the problem is solved. In SageMath9.0, I need to use Matrix instead of matrix. Thank you very much.