Through this tutorial, I want to explain how to compuate the multiplications of non-square matrix; for example, the result of multiplying a $2 \times 5$ matrix by a $5 \times 2$ matrix (a by c in the following example.)
import numpy as np
a = np.array([[1, 2,7,9,10], [3, 4,5,12,11]])
a
array([[ 1, 2, 7, 9, 10],
[ 3, 4, 5, 12, 11]])
a is a $2 \times 5$ matrix. It means the transformation from a 5d space to a 2d space. You can consider each column of a as the corresponding coordinates of each of the five axes (of the original 5d space).
Suppose we have a vector in a 5d space, i.e., a $5 \times 1$ vector:
b= np.array([1,2,3,4,5]).reshape(-1, 1)
b
array([[1],
[2],
[3],
[4],
[5]])
Multiplying a by b means mapping b onto a. The resulting $2 \times 1$ matrix is the result vector.
a @ b
array([[112],
[129]])
Matrix multiplication #
Suppose we have a $5 \times 2$ matrix:
c = np.array([[1,2], [3,4], [4,7], [9,10], [11,-1]])
c
array([[ 1, 2],
[ 3, 4],
[ 4, 7],
[ 9, 10],
[11, -1]])
c means transforming a 2d space to 5d. A 2d space has two axes ($\vec{x}$ and $\vec{y}$). The first column of c is the coordinates of $\vec{x}$ in the new 5d space, and the second column, $\vec{y}$.
What does $c \times a$ means? This matrix multiplication is the result of two space transformations: first a and then c. That is to say, it means we first transform a 5d space to 2d and then transform 2d space to 5d.
To understand what $c \times a$ means, let’s first consider multiplying a $2 \times 1$ matrix, i.e., a vector in two-dimensional space, by c:
d = np.array([[1], [2]])
d
array([[1],
[2]])
$\vec{d}$ means $1 \vec{x} + 2 \vec{y}$. Since in the new 5d space, the coordinates of $\vec{x}$ is [1, 3, 4, 9, 11] and the coordinates of $\vec{y}$ is [2, 4, 7, 10, -1], the result of $c \times d$ will be:
c @ d
array([[ 5],
[11],
[18],
[29],
[ 9]])
That is the coordinates of $\vec{d}$ in the new 5d space.
The calculation of $c \times a$ is the same. You can view each column of a as a $\vec{d}$.
e = c @ a
e
array([[ 7, 10, 17, 33, 32],
[ 15, 22, 41, 75, 74],
[ 25, 36, 63, 120, 117],
[ 39, 58, 113, 201, 200],
[ 8, 18, 72, 87, 99]])
For any vector in a five-dimensional space, say f = [1, 5, -9, 9, 8], multiplying it by a and then by c is the same as multiplying it by e directly.
f = np.array([[1], [5], [-9], [9], [8]])
g = a @ f
h = c @ g
h
array([[ 457],
[1023],
[1654],
[2721],
[1025]])
e @ f
array([[ 457],
[1023],
[1654],
[2721],
[1025]])
Last modified on 2022-12-12