Linear algebra

Elementwise operations are not always what we expect. We have to distinguish between a two-dimensional array and the interpretation of it as a matrix. For example, the product of two matrices is not the element-wise product, but the product of the rows of the first matrix with the columns of the second matrix. This is why there are special operators for linear algebra.

np.arange(4) * np.arange(4)          # Elementwise operation: array([0, 1, 4, 9])
np.dot(np.arange(4), np.arange(4))   # Scalar product:          14
np.outer(np.arange(4), np.arange(4)) # Outer product:        array([[0, 0, 0, 0],
                                     #                                [0, 1, 2, 3],
                                     #                                [0, 2, 4, 6],
                                     #                                [0, 3, 6, 9]])

The matrix multiplication is performed with @:

A = np.arange(9).reshape(3,3) #                         array([[0, 1, 2],
                              #                                [3, 4, 5],
                              #                                [6, 7, 8]])
A * A                         # Elementwise operation: array([[ 0,  1,  4],
                              #                                [ 9, 16, 25],
                              #                                [36, 49, 64]])
A @ A                         # Matrix multiplication:  array([[ 15,  18,  21],
                              #                                [ 42,  54,  66],
                              #                                [ 69,  90, 111]])

The norm of a vector can be calculated with np.linalg.norm:

np.linalg.norm(np.arange(3))

Eigenvalues and eigenvectors as well as the determinant of a matrix can be calculated with np.linalg.eig:

A = np.array([[2,1], [1,2]])
np.linalg.eig(A).eigenvalues  # array([3., 1.])
np.linalg.eig(A).eigenvectors # array([[ 0.70710678, -0.70710678],
                              #        [ 0.70710678,  0.70710678]])
np.linalg.det(A)              # 2.9999999999999996