Two-Dimensional Numpy

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Basic 2-dimensional matrix representation

Numpy arrays can be N-Dimensional, and are represented as matrices instead of nested lists.

a = [[11,12,13],[21,22,23],[31,32,33]]

A = np.array(a)

$$ A: \begin{bmatrix} 11 & 12 & 13 \ 21 & 22 & 23 \ 31 & 32 & 33 \ \end{bmatrix} $$

A.ndim: 2 because the depth of list nesting, or the dimensionality of the array, is two levels or ranks, so

A.shape returns a tuple (3,3) representing a first rank of 3 elements and a second rank of 3 elements forming the 'shape' of the matrix.

A.size: 9 represents the total number of elements in the matrix, shorthanded by multiplying the elements returned in the shape tuple.

Matrix axis labelling

By convention, in rectangular notation, the vertical axis is axis 0, while the horizontal is axis 1. This corresponds to the nesting depth being axes 0 and the element depth being axis 1. Were a third dimension to be added, axis 2 would represent the element complexity.

Slicing in N-Dimensional arrays

This is fairly easy conceptually and syntactically.

The comma tells you which dimension you're slicing into with the colon. So let's take the matrix from before.

$$ A: \begin{bmatrix} 11 & 12 & 13 \ 21 & 22 & 23 \ 31 & 32 & 33 \ \end{bmatrix} $$

Let's slice it up a bit.

$$ A[0:2,1]: \begin{bmatrix} 11 & [12] & 13 \ 21 & [22] & 23 \ 31 & 32 & 33 \ \end{bmatrix} \quad A[1,1:2]: \begin{bmatrix} 11 & 12 & 13 \ 21 & [22 & 23] \ 31 & 32 & 33 \ \end{bmatrix} \quad A[0:2,0:2]: \begin{bmatrix} [11 & 12] & 13 \ [21 & 22] & 23 \ 31 & 32 & 33 \ \end{bmatrix} $$

Matrix operations

Addition

$$ X = \begin{bmatrix} 1 & 0 \ 0 & 1 \ \end{bmatrix} \quad Y = \begin{bmatrix} 2 & 1 \ 1 & 2 \ \end{bmatrix} $$ $$ X+Y= \begin{bmatrix} 1+2 & 1+1 \ 0+1 & 1+2 \ \end{bmatrix}

\begin{bmatrix} 3 & 2 \ 1 & 3 \ \end{bmatrix} $$

X = np.array([1,0],[0,1])
Y = np.array([2,1],[1,2])
Z = X + Y
Z: ([3,2],[1,3])

Multiplication (Hadamar product)

Note: This only works for identically-shaped matrices. See below for the rules on multiplication of matrices.

$$ X = \begin{bmatrix} 1 & 0 \ 0 & 1 \ \end{bmatrix} \quad Y = \begin{bmatrix} 2 & 1 \ 1 & 2 \ \end{bmatrix} $$ $$ X \circ Y= \begin{bmatrix} (1)2 & (1)1 \ (0)1 & (1)2 \ \end{bmatrix}

\begin{bmatrix} 2 & 1 \ 0 & 2 \ \end{bmatrix} $$

X = np.array([1,0],[0,1])
Y = np.array([2,1],[1,2])
Z = X * Y
Z: ([2,1],[0,2])

Adding and Multiplying by scalars

Addition

$$ Y = \begin{bmatrix} 2 & 1 \ 1 & 2 \ \end{bmatrix} \quad 2+Y = \begin{bmatrix} 2+2 & 2+1 \ 2+1 & 2+2 \ \end{bmatrix}

\begin{bmatrix} 4 & 3 \ 3 & 4 \ \end{bmatrix} $$

Y = np.array([2,1],[1,2])
Z = 2 + Y
Z: ([4,3],[3,4])

Multiplication

$$ Y = \begin{bmatrix} 2 & 1 \ 1 & 2 \ \end{bmatrix} \quad 2Y = \begin{bmatrix} 22 & 21 \ 21 & 22 \ \end{bmatrix}

\begin{bmatrix} 4 & 2 \ 2 & 4 \ \end{bmatrix} $$

Y = np.array([2,1],[1,2])
Z = 2Y
Z: ([4,2],[2,4])

Matrix Multiplication

When the shape of two matrices is different, one must have the same number of columns as the other has rows for them to be able to multiply.

If this rule is met, multiplication is handled by applying the:

• dot product of the top row of the first matrix to the first column of the second matrix
• dot product of the top row of the first matrix to the second column of the second matrix
• dot product of the bottom row of the first matrix to the first column of the second matrix
• dot product of the bottom row of the first matrix to the second column of the second matrix

$$ A= \begin{bmatrix} 0 & 1 & 1 \ 1 & 0 & 1 \ \end{bmatrix} \quad \quad B= \begin{bmatrix} 1 & 1 \ 1 & 1 \ -1 & 1 \ \end{bmatrix} $$

0 * 1 + 1 * 1 + 1 * -1 = 0
0 * 1 + 1 * 1 + 1 * 1 = 2
1 * 1 + 0 * 1 + 1 * -1 = 0 
1 * 1 + 0 * 1 + 1 * 1 = 2 

$$ AB= \begin{bmatrix} 0 & 2 \ 0 & 2 \ \end{bmatrix} $$