TopolearnSymbolic linear algebra gives exact closed forms for small systems, Jacobians and parameterised models.
Matrix([[1, 2], [3, 4]]) creates a matrix; entries can be numbers, symbols or any expression. Helper constructors include eye(n), zeros(m, n), ones(m, n) and diag(...). A flat list gives a column vector.
Operators follow linear algebra: A * B and A @ B are matrix products, A**-1 or A.inv() the inverse, A.T the transpose. The most used methods are:
A.det(),A.rank(),A.rref()(reduced row echelon form plus pivot columns),A.eigenvals()(dict of eigenvalue to multiplicity) andA.eigenvects(),A.nullspace(),A.columnspace(),A.LUsolve(b)orA.solve(b)forA x = b,A.subs(...),A.applyfunc(simplify)to work entry-wise,Matrix(fs).jacobian(vars)for Jacobians.
Everything is exact, which is the point and the cost. Symbolic determinants and inverses grow explosively with size; a 6x6 fully symbolic inverse can take a very long time and produce enormous output. For numeric matrices beyond a handful of rows, use NumPy.
Matrix is mutable (you can assign A[0, 1] = 5), which also means it is not hashable. Use ImmutableMatrix when you need to put a matrix inside other expressions or use it as a dict key.
Example
from sympy import Matrix, symbols, eye, simplify
a = symbols("a")
A = Matrix([[1, 2], [3, 4]])
print(A * A) # Matrix([[7, 10], [15, 22]])
print(A.inv()) # Matrix([[-2, 1], [3/2, -1/2]])
print(A.det()) # -2
M = Matrix([[a, 1], [1, a]])
print(M.eigenvals()) # {a - 1: 1, a + 1: 1}
print(M.det()) # a**2 - 1
print(Matrix([[1, 2, 3], [2, 4, 6]]).rref()) # (Matrix([[1, 2, 3], [0, 0, 0]]), (0,))
b = Matrix([5, 11])
print(A.solve(b)) # Matrix([[1], [2]])
x, y = symbols("x y")
print(Matrix([x**2 * y, x + y]).jacobian([x, y])) # Matrix([[2*x*y, x**2], [1, 1]])Remember
- Matrix entries can be any expression; * and @ are matrix products
- det, inv, rref, eigenvals, solve, jacobian cover most needs
- Symbolic cost explodes with size; use NumPy for big numeric matrices
Watch out for
- Trying to invert a large fully symbolic matrix and waiting forever.