def diagonalize(A):
assert A.ndim == 2
assert A.shape[0] == A.shape[1]
eigenvalues, eigenvectors = eig(A)
P = eigenvectors
if rank(Matrix(P.tolist())) != A.shape[0]:
print("Matrix can not be diagonalized!")
return None, None, None
D = np.diag(eigenvalues)
Pinv = inv(P)
return P, D, Pinv
def gram_schmidt_process(basis):
matrix = Matrix(basis)
assert rank(matrix) == len(basis)
res = [basis[0]]
for i in range(1, len(basis)):
p = basis[i]
for r in res:
p = p - basis[i].dot(r) / r.dot(r) * r
res.append(p)
return res
[-1, 1, 0, 2, -5],
[1, -1, 4, 2, 4],
[-2, 2, -5, -1, -3]])
b2 = Vector([1, 5, 13, -1])
ls2 = LinearSystem(A2, b2)
ls2.gauss_jordan_elimination()
ls2.fancy_print()
print()
A3 = Matrix([[2, 2],
[2, 1],
[1, 2]])
b3 = Vector([3, 2.5, 7])
ls3 = LinearSystem(A3, b3)
if not ls3.gauss_jordan_elimination():
print("No solution")
ls3.fancy_print()
print()
A = Matrix([[1, 2], [3, 4]])
invA = inv(A)
print(invA)
print(invA.dot(A))
print(A.dot(invA))
print(rank(A1))
b6 = Vector([3, -1, 8])
ls6 = LinearSystem(A6, b6)
ls6.gauss_jordan_elimination()
ls6.fancy_print()
print()
# [1, 1, 1]
A7 = Matrix([[1, -1, 2, 0, 3], [-1, 1, 0, 2, -5], [1, -1, 4, 2, 4],
[-2, 2, -5, -1, -3]])
b7 = Vector([1, 5, 13, -1])
ls7 = LinearSystem(A7, b7)
ls7.gauss_jordan_elimination()
ls7.fancy_print()
print()
A8 = Matrix([[2, 2], [2, 1], [1, 2]])
b8 = Vector([3, 2.5, 7])
ls8 = LinearSystem(A8, b8)
if not ls8.gauss_jordan_elimination():
print("No Solution!")
ls8.fancy_print()
print()
A = Matrix([[1, 2], [3, 4]])
invA = inv(A)
print(invA)
print(A.dot(invA))
print(invA.dot(A))
print(rank(A))
print(rank(A7))
if __name__ == '__main__':
A = Matrix([[1, 2, 4], [3, 7, 2], [2, 3, 3]])
b = Vector([7, -11, 1])
ls = LinearSystem(A, b)
ls.gauss_jordan_elimination()
ls.fancy_print()
print()
A7 = Matrix([[1, -1, 2, 0, 3], [-1, 1, 0, 2, -5], [1, -1, 4, 2, 4],
[-2, 2, -5, -1, -3]])
b7 = Vector([1, 5, 13, -1])
ls7 = LinearSystem(A7, b7)
ls7.gauss_jordan_elimination()
ls7.fancy_print()
print()
A8 = Matrix([[2, 2], [2, 1], [1, 2]])
b8 = Vector([3, 2.5, 7])
ls8 = LinearSystem(A8, b8)
if not ls8.gauss_jordan_elimination():
print("No Solution!")
ls8.fancy_print()
A = Matrix([[1, 2], [3, 4]])
invA = inv(A)
print(invA)
print(A.dot(invA))
print(invA.dot(A))
print("rank A8 = {}".format(rank(A8)))
ls6.fancy_print()
print()
# [1, 1, 1]
A7 = Matrix([[1, -1, 2, 0, 3], [-1, 1, 0, 2, -5], [1, -1, 4, 2, 4],
[-2, 2, -5, -1, -3]])
b7 = Vector([1, 5, 13, -1])
ls7 = LinearSystem(A7, b7)
ls7.gauss_jordan_elimination()
ls7.fancy_print()
print()
# [-15, 5, 2, 0]
A8 = Matrix([[2, 2], [2, 1], [1, 2]])
b8 = Vector([3, 2.5, 7])
ls8 = LinearSystem(A8, b8)
if not ls8.gauss_jordan_elimination():
print("No Solution!")
ls8.fancy_print()
print()
A = Matrix([[1, 2], [3, 4]])
invA = inv(A)
print(invA)
print(A.dot(invA))
print(invA.dot(A))
print()
print("rank A8 = {}".format(rank(A8)))
print("rank A7 = {}".format(rank(A7)))
print("rank A6 = {}".format(rank(A6)))
ls2 = LinearSystem(A2, b2)
ls2.gauss_jordan_elimination()
ls2.fancy_print()
print()
A3 = Matrix([[1, -1, 2, 0, 3], [-1, 1, 0, 2, -5], [1, -1, 4, 2, 4],
[-2, 2, -5, -1, -3]])
b3 = Vector([1, 5, 13, -1])
ls3 = LinearSystem(A3, b3)
if not ls3.gauss_jordan_elimination():
print("No solution")
print("ls3")
ls3.fancy_print()
print()
A4 = Matrix([[2, 2], [2, 1], [1, 2]])
b4 = Vector([3, 2.5, 7])
ls4 = LinearSystem(A4, b4)
if not ls4.gauss_jordan_elimination():
print("No solution")
ls4.fancy_print()
print()
A5 = Matrix([[1, 2], [3, 4]])
invA5 = inv(A5)
print(invA5)
print(A5.dot(invA5))
print(invA5.dot(A5))
print(rank(A3))
b = Vector([7, -11, 1])
ls = LinearSystem(A, b)
ls.gauss_jordan_elimination()
ls.fancy_print()
print()
A7 = Matrix([[1, -1, 2, 0, 3], [-1, 1, 0, 2, -5], [1, -1, 4, 2, 4],
[-2, 2, -5, -1, -3]])
b7 = Vector([1, 5, 13, -1])
ls7 = LinearSystem(A7, b7)
ls7.gauss_jordan_elimination()
ls7.fancy_print()
print()
A8 = Matrix([[2, 2], [2, 1], [1, 2]])
b8 = Vector([3, 2.5, 7])
ls8 = LinearSystem(A8, b8)
if not ls8.gauss_jordan_elimination():
print("no solution!")
ls8.fancy_print()
print()
A9 = Matrix([[1, 2], [3, 4]])
invA = inv(A9)
print(invA)
print(invA.row_num(), '+', invA.col_num())
print(A9.dot(invA))
print
print(rank(A8))
