def test_Identity():
A = MatrixSymbol('A', n, m)
i, j = symbols('i j')
In = Identity(n)
Im = Identity(m)
assert A * Im == A
assert In * A == A
assert transpose(In) == In
assert In.inverse() == In
assert In.conjugate() == In
assert In[i, j] != 0
assert Sum(In[i, j], (i, 0, n - 1), (j, 0, n - 1)).subs(n, 3).doit() == 3
assert Sum(Sum(In[i, j], (i, 0, n - 1)), (j, 0, n - 1)).subs(n,
3).doit() == 3
# If range exceeds the limit `(0, n-1)`, do not remove `Piecewise`:
expr = Sum(In[i, j], (i, 0, n - 1))
assert expr.doit() == 1
expr = Sum(In[i, j], (i, 0, n - 2))
assert expr.doit().dummy_eq(
Piecewise((1, (j >= 0) & (j <= n - 2)), (0, True)))
expr = Sum(In[i, j], (i, 1, n - 1))
assert expr.doit().dummy_eq(
Piecewise((1, (j >= 1) & (j <= n - 1)), (0, True)))
assert Identity(3).as_explicit() == ImmutableMatrix.eye(3)
