import random import pytest from sympy.core.numbers import I from sympy.core.numbers import Rational from sympy.core.symbol import (Symbol, symbols) from sympy.functions.elementary.miscellaneous import sqrt from sympy.polys.polytools import Poly from sympy.matrices import Matrix, eye, ones from sympy.abc import x, y, z from sympy.testing.pytest import raises from sympy.matrices.exceptions import NonSquareMatrixError from sympy.functions.combinatorial.factorials import factorial, subfactorial @pytest.mark.parametrize("method", [ # Evaluating these directly because they are never reached via M.det() Matrix._eval_det_bareiss, Matrix._eval_det_berkowitz, Matrix._eval_det_bird, Matrix._eval_det_laplace, Matrix._eval_det_lu ]) @pytest.mark.parametrize("M, sol", [ (Matrix(), 1), (Matrix([[0]]), 0), (Matrix([[5]]), 5), ]) def test_eval_determinant(method, M, sol): assert method(M) == sol @pytest.mark.parametrize("method", [ "domain-ge", "bareiss", "berkowitz", "bird", "laplace", "lu"]) @pytest.mark.parametrize("M, sol", [ (Matrix(( (-3, 2), ( 8, -5) )), -1), (Matrix(( (x, 1), (y, 2*y) )), 2*x*y - y), (Matrix(( (1, 1, 1), (1, 2, 3), (1, 3, 6) )), 1), (Matrix(( ( 3, -2, 0, 5), (-2, 1, -2, 2), ( 0, -2, 5, 0), ( 5, 0, 3, 4) )), -289), (Matrix(( ( 1, 2, 3, 4), ( 5, 6, 7, 8), ( 9, 10, 11, 12), (13, 14, 15, 16) )), 0), (Matrix(( (3, 2, 0, 0, 0), (0, 3, 2, 0, 0), (0, 0, 3, 2, 0), (0, 0, 0, 3, 2), (2, 0, 0, 0, 3) )), 275), (Matrix(( ( 3, 0, 0, 0), (-2, 1, 0, 0), ( 0, -2, 5, 0), ( 5, 0, 3, 4) )), 60), (Matrix(( ( 1, 0, 0, 0), ( 5, 0, 0, 0), ( 9, 10, 11, 0), (13, 14, 15, 16) )), 0), (Matrix(( (3, 2, 0, 0, 0), (0, 3, 2, 0, 0), (0, 0, 3, 2, 0), (0, 0, 0, 3, 2), (0, 0, 0, 0, 3) )), 243), (Matrix(( (1, 0, 1, 2, 12), (2, 0, 1, 1, 4), (2, 1, 1, -1, 3), (3, 2, -1, 1, 8), (1, 1, 1, 0, 6) )), -55), (Matrix(( (-5, 2, 3, 4, 5), ( 1, -4, 3, 4, 5), ( 1, 2, -3, 4, 5), ( 1, 2, 3, -2, 5), ( 1, 2, 3, 4, -1) )), 11664), (Matrix(( ( 2, 7, -1, 3, 2), ( 0, 0, 1, 0, 1), (-2, 0, 7, 0, 2), (-3, -2, 4, 5, 3), ( 1, 0, 0, 0, 1) )), 123), (Matrix(( (x, y, z), (1, 0, 0), (y, z, x) )), z**2 - x*y), ]) def test_determinant(method, M, sol): assert M.det(method=method) == sol def test_issue_13835(): a = symbols('a') M = lambda n: Matrix([[i + a*j for i in range(n)] for j in range(n)]) assert M(5).det() == 0 assert M(6).det() == 0 assert M(7).det() == 0 def test_issue_14517(): M = Matrix([ [ 0, 10*I, 10*I, 0], [10*I, 0, 0, 10*I], [10*I, 0, 5 + 2*I, 10*I], [ 0, 10*I, 10*I, 5 + 2*I]]) ev = M.eigenvals() # test one random eigenvalue, the computation is a little slow test_ev = random.choice(list(ev.keys())) assert (M - test_ev*eye(4)).det() == 0 @pytest.mark.parametrize("method", [ "bareis", "det_lu", "det_LU", "Bareis", "BAREISS", "BERKOWITZ", "LU"]) @pytest.mark.parametrize("M, sol", [ (Matrix(( ( 3, -2, 0, 5), (-2, 1, -2, 2), ( 0, -2, 5, 0), ( 5, 0, 3, 4) )), -289), (Matrix(( (-5, 2, 3, 4, 5), ( 1, -4, 3, 4, 5), ( 1, 2, -3, 4, 5), ( 1, 2, 3, -2, 5), ( 1, 2, 3, 4, -1) )), 11664), ]) def test_legacy_det(method, M, sol): # Minimal support for legacy keys for 'method' in det() # Partially copied from test_determinant() assert M.det(method=method) == sol def eye_Determinant(n): return Matrix(n, n, lambda i, j: int(i == j)) def zeros_Determinant(n): return Matrix(n, n, lambda i, j: 0) def test_det(): a = Matrix(2, 3, [1, 2, 3, 4, 5, 6]) raises(NonSquareMatrixError, lambda: a.det()) z = zeros_Determinant(2) ey = eye_Determinant(2) assert z.det() == 0 assert ey.det() == 1 x = Symbol('x') a = Matrix(0, 0, []) b = Matrix(1, 1, [5]) c = Matrix(2, 2, [1, 2, 3, 4]) d = Matrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 8]) e = Matrix(4, 4, [x, 1, 2, 3, 4, 5, 6, 7, 2, 9, 10, 11, 12, 13, 14, 14]) from sympy.abc import i, j, k, l, m, n f = Matrix(3, 3, [i, l, m, 0, j, n, 0, 0, k]) g = Matrix(3, 3, [i, 0, 0, l, j, 0, m, n, k]) h = Matrix(3, 3, [x**3, 0, 0, i, x**-1, 0, j, k, x**-2]) # the method keyword for `det` doesn't kick in until 4x4 matrices, # so there is no need to test all methods on smaller ones assert a.det() == 1 assert b.det() == 5 assert c.det() == -2 assert d.det() == 3 assert e.det() == 4*x - 24 assert e.det(method="domain-ge") == 4*x - 24 assert e.det(method='bareiss') == 4*x - 24 assert e.det(method='berkowitz') == 4*x - 24 assert f.det() == i*j*k assert g.det() == i*j*k assert h.det() == 1 raises(ValueError, lambda: e.det(iszerofunc="test")) def test_permanent(): M = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]]) assert M.per() == 450 for i in range(1, 12): assert ones(i, i).per() == ones(i, i).T.per() == factorial(i) assert (ones(i, i)-eye(i)).per() == (ones(i, i)-eye(i)).T.per() == subfactorial(i) a1, a2, a3, a4, a5 = symbols('a_1 a_2 a_3 a_4 a_5') M = Matrix([a1, a2, a3, a4, a5]) assert M.per() == M.T.per() == a1 + a2 + a3 + a4 + a5 def test_adjugate(): x = Symbol('x') e = Matrix(4, 4, [x, 1, 2, 3, 4, 5, 6, 7, 2, 9, 10, 11, 12, 13, 14, 14]) adj = Matrix([ [ 4, -8, 4, 0], [ 76, -14*x - 68, 14*x - 8, -4*x + 24], [-122, 17*x + 142, -21*x + 4, 8*x - 48], [ 48, -4*x - 72, 8*x, -4*x + 24]]) assert e.adjugate() == adj assert e.adjugate(method='bareiss') == adj assert e.adjugate(method='berkowitz') == adj assert e.adjugate(method='bird') == adj assert e.adjugate(method='laplace') == adj a = Matrix(2, 3, [1, 2, 3, 4, 5, 6]) raises(NonSquareMatrixError, lambda: a.adjugate()) def test_util(): R = Rational v1 = Matrix(1, 3, [1, 2, 3]) v2 = Matrix(1, 3, [3, 4, 5]) assert v1.norm() == sqrt(14) assert v1.project(v2) == Matrix(1, 3, [R(39)/25, R(52)/25, R(13)/5]) assert Matrix.zeros(1, 2) == Matrix(1, 2, [0, 0]) assert ones(1, 2) == Matrix(1, 2, [1, 1]) assert v1.copy() == v1 # cofactor assert eye(3) == eye(3).cofactor_matrix() test = Matrix([[1, 3, 2], [2, 6, 3], [2, 3, 6]]) assert test.cofactor_matrix() == \ Matrix([[27, -6, -6], [-12, 2, 3], [-3, 1, 0]]) test = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]]) assert test.cofactor_matrix() == \ Matrix([[-3, 6, -3], [6, -12, 6], [-3, 6, -3]]) def test_cofactor_and_minors(): x = Symbol('x') e = Matrix(4, 4, [x, 1, 2, 3, 4, 5, 6, 7, 2, 9, 10, 11, 12, 13, 14, 14]) m = Matrix([ [ x, 1, 3], [ 2, 9, 11], [12, 13, 14]]) cm = Matrix([ [ 4, 76, -122, 48], [-8, -14*x - 68, 17*x + 142, -4*x - 72], [ 4, 14*x - 8, -21*x + 4, 8*x], [ 0, -4*x + 24, 8*x - 48, -4*x + 24]]) sub = Matrix([ [x, 1, 2], [4, 5, 6], [2, 9, 10]]) assert e.minor_submatrix(1, 2) == m assert e.minor_submatrix(-1, -1) == sub assert e.minor(1, 2) == -17*x - 142 assert e.cofactor(1, 2) == 17*x + 142 assert e.cofactor_matrix() == cm assert e.cofactor_matrix(method="bareiss") == cm assert e.cofactor_matrix(method="berkowitz") == cm assert e.cofactor_matrix(method="bird") == cm assert e.cofactor_matrix(method="laplace") == cm raises(ValueError, lambda: e.cofactor(4, 5)) raises(ValueError, lambda: e.minor(4, 5)) raises(ValueError, lambda: e.minor_submatrix(4, 5)) a = Matrix(2, 3, [1, 2, 3, 4, 5, 6]) assert a.minor_submatrix(0, 0) == Matrix([[5, 6]]) raises(ValueError, lambda: Matrix(0, 0, []).minor_submatrix(0, 0)) raises(NonSquareMatrixError, lambda: a.cofactor(0, 0)) raises(NonSquareMatrixError, lambda: a.minor(0, 0)) raises(NonSquareMatrixError, lambda: a.cofactor_matrix()) def test_charpoly(): x, y = Symbol('x'), Symbol('y') z, t = Symbol('z'), Symbol('t') from sympy.abc import a,b,c m = Matrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 9]) assert eye_Determinant(3).charpoly(x) == Poly((x - 1)**3, x) assert eye_Determinant(3).charpoly(y) == Poly((y - 1)**3, y) assert m.charpoly() == Poly(x**3 - 15*x**2 - 18*x, x) raises(NonSquareMatrixError, lambda: Matrix([[1], [2]]).charpoly()) n = Matrix(4, 4, [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]) assert n.charpoly() == Poly(x**4, x) n = Matrix(4, 4, [45, 0, 0, 0, 0, 23, 0, 0, 0, 0, 87, 0, 0, 0, 0, 12]) assert n.charpoly() == Poly(x**4 - 167*x**3 + 8811*x**2 - 173457*x + 1080540, x) n = Matrix(3, 3, [x, 0, 0, a, y, 0, b, c, z]) assert n.charpoly() == Poly(t**3 - (x+y+z)*t**2 + t*(x*y+y*z+x*z) - x*y*z, t)