Linear Algebra using NumPy

For beginner NumPy users with a linear algebra background, it’s easy to understand that to define a matrix, $A=\begin{bmatrix}1&2\\3&4\end{bmatrix}$, we should use

A = np.array([[1,2],[3,4]])

since

    A
    Out[2]:
    array([[1, 2],
           [3, 4]])
    A.shape
    Out[3]: (2, 2)

However, how should $x=\begin{bmatrix}1\\2\end{bmatrix}$ be defined? Should we use a 1-D array

    x1 = np.array([1,2])
    x1
    Out[4]: array([1, 2])
    x1.shape
    Out[5]: (2,)

or a 2-D array with 1 column?

    x2=np.array([[1],[2]])
    x2
    Out[6]:
    array([[1],
           [2]])
    x2.shape
    Out[7]: (2, 1)

If we were to decide based on the output of the console, x2 would seem to be the most appropriate since it looks like a column vector in the console, while x1 looks like a row vector. It turns out though, that the correct way is to use x1 to represent $x$. A vector (in the linear algebra sense) should be represented by a 1-D array in NumPy. A matrix (in the linear algebra sense) should be represented by a 2-D array in NumPy.

In fact, for NumPy, there’s no difference between a row or a column vector when it is represented by a 1-D array. The @ operator overloads matrix-vector and vector-matrix multiplication so the practitioner does not need to worry about transposing vectors. For example, to compute $x^TA=\begin{bmatrix}7&10\end{bmatrix}$, use

    x1 @ A
    Out[8]: array([ 7, 10])

and to compute $Ax=\begin{bmatrix}5\\11\end{bmatrix}$, use

    A @ x1
    Out[9]: array([ 5, 11])

Also, the weighted scalar product $x^TAx=27$ can be computed by

    x1 @ A @ x1
    Out[10]: 27

How about the outer product, $xx^T=\begin{bmatrix}1&2\\2&4\end{bmatrix}$? This computation does require something beyond the @ operator, namely, using NumPy’s outer function

    np.outer(x1,x1)
    Out[11]:
    array([[1, 2],
           [2, 4]])