def test_Matrix():
M = Matrix.id(3)
assert M.rank() == 3
assert M.is_sim(M.inv())
assert M.is_sim(M.pinv())
assert M.is_diagonal()
assert not M.is_null()
X = Matrix.from_str("1 1 0; 0 1 0; 0 0 0")
Y = Matrix.from_str("1 -1 0; 0 1 0; 0 0 0")
assert X.rank() == 2
assert X.pinv().is_sim(Y)
Z = Matrix.from_str("1 -1 0; 0 1 0; 0 0 0; 0 0 0")
W = Matrix.from_str("1 -1 0 1; 0 1 0 1; 0 0 0 1")
assert Y.join_row(Matrix([0,0,0]).to_row()).is_sim(Z)
assert Y.join_col(Matrix([1,1,1]).to_col()).is_sim(W)
P = M @ M.pinv()
Q = M.pinv() @ M
assert P.is_sim(P @ P)
assert Q.is_sim(Q @ Q)
@given(hnp.arrays(np.float64, (3,3)))
def test_rand_float_matrix(A):
assume(np.isfinite(A).all())
M = Matrix(A)
P = M @ M.pinv()
Q = M.pinv() @ M
assert P.is_sim(P @ P)
assert Q.is_sim(Q @ Q)
@given(hnp.arrays(np.float64, (3,3), elements=st.floats(0,1)))
@example(Matrix.from_str("0 0 0"))
@example(Matrix.from_str("0 0 0"))
@example(Matrix.from_str("1 1 1 ; 3 1 5"))
@example(Matrix.from_str("0 1 ; 1 1"))
@example(Matrix.rand_rk(5,5,3))
@example(Matrix.rand_rk(3,5,3))
@example(Matrix.rand_rk(5,3,3))
@example(Matrix.rand_rk(30,20,3))
def test_rd_lu(A):
A = Matrix(A)
assume(np.isfinite(A).all())
res = A.rank_decomposition(method="lu")
assert res.is_valid()
assert res.R.is_diagonal()
assert (res.map(A) - res.R).is_null()
assert res.R.rank() == A.rank()