Matrix Transformations

Matrix transformation or linear transformation of a vector v is pre-multiplying the vector by a matrix (called the transformation matrix) which results in v being transformed.

Let us first interpret matrix transformations in the 2D space.

from sympy import Matrix, MatrixSymbol, Eq, MatMul

sympy_v = MatrixSymbol("v",2,1) # Column vector having 2 rows
sympy_A = MatrixSymbol("A",2,2) # 2x2 Transformation matrix
sympy_y = MatrixSymbol("y", 2, 1)
print("y is the transformed vector.")
Eq(sympy_y,sympy_A*sympy_v,evaluate=False)

y is the transformed vector.

\(\displaystyle y = A v\)

Example

import numpy as np
import matplotlib.pyplot as plt

A = np.array([[3,3],[2,6]])
v = np.array([1,1])
Av = A @ v
plt.arrow(0,0,1,1,head_width=0.2,width=0.05,color="blue",label="v")
plt.arrow(0,0,Av[0],Av[1],head_width=0.2,width=0.05,color='red',label="Av")
plt.legend([f"v = {v}",f"Av = {Av}"],loc='lower right')
plt.title(f"A = {A}")
plt.show()

Eq(Matrix(Av),MatMul(Matrix(A),Matrix(v)),evaluate=False)

\(\displaystyle \left[\begin{matrix}6\\8\end{matrix}\right] = \left[\begin{matrix}3 & 3\\2 & 6\end{matrix}\right] \left[\begin{matrix}1\\1\end{matrix}\right]\)

In the above equation,

A = \(\begin{bmatrix} 3&3\\2&6\end{bmatrix}\), v = \(\begin{bmatrix} 1\\1 \end{bmatrix}\) and y = \(\begin{bmatrix} 6\\8 \end{bmatrix}\)

Note:

If the transformation matrix A is taken to be \(\begin{bmatrix} \cos(\theta)&-\sin(\theta)\\ \sin(\theta)&\cos(\theta) \end{bmatrix}\)

then the vector v is rotated in an anti-clockwise direction by an angle \(\theta\).

For example, Let’s take v = \(\begin{bmatrix} 1\\0 \end{bmatrix}\) and \(\theta\)=90 deg, therefore A = \(\begin{bmatrix} 0&-1\\1&0 \end{bmatrix}\)

A = np.array([[0,-1],[1,0]])
v = np.array([1,0])
Av = A @ v
plt.figure()
plt.arrow(0,0,1,0,head_width=0.1,width=0.01,color="blue",label="v")
plt.arrow(0,0,Av[0],Av[1],head_width=0.1,width=0.01,color='red',label="Av")
plt.legend([f"v = {v}",f"Av = {Av}"])
plt.title(f"A = {A}")
plt.show()

Eq(Matrix(Av),MatMul(Matrix(A),Matrix(v)),evaluate=False)

\(\displaystyle \left[\begin{matrix}0\\1\end{matrix}\right] = \left[\begin{matrix}0 & -1\\1 & 0\end{matrix}\right] \left[\begin{matrix}1\\0\end{matrix}\right]\)

As a result of the transformation, vector v is rotated by 90 degrees in the counter-clockwise direction.

Understanding Matrix transformations of low-rank matrices

Low-rank matrices are matrices which have rank less than the dimensionality of the column space of the matrix. For eg. a 3x3 transformation matrix will be a low rank matrix if it’s column space can be spanned by less than or equal to 2 vectors. In other words, the column vectors are linearly dependant.

Example of a low-rank matrix is \(\begin{bmatrix} 1&0&1\\0.5&1&1.5\\1&0&1 \end{bmatrix}\)

The determinant of the transformation matrix having low-rank is zero.

Geometrically what this means is transforming the vector using a low-rank matrix results in the subspace of the original vector space. Hence for example if the original vector space is \(R^3\), the resultant subspace can be \(R^2\), \(R\), or in an extreme case even a point.

Example

A = np.array([[1,-1],[3,-3]]) # Low-rank 2x2 matrix
# Example 1
v = np.array([2,3])
Av = A @ v
plt.figure()
plt.subplot(1,2,1)
plt.arrow(0,0,2,3,head_width=0.1,width=0.01,color="blue",label="v")
plt.arrow(0,0,Av[0],Av[1],head_width=0.1,width=0.01,color='red',label="Av")
plt.title(f"A = {A}")
x = np.linspace(-3,3)
y = x*3
plt.plot(x,y,color="black",alpha=0.3)
plt.legend([f"v = {v}",f"Av = {Av}","y=3x"])

# Example 2
v1 = np.array([-1.7,2.2])
plt.subplot(1,2,2)
plt.arrow(0,0,-1.7,2.2,head_width=0.1,width=0.01,color="blue",label="v")
plt.arrow(0,0,Av[0],Av[1],head_width=0.1,width=0.01,color='red',label="Av")
plt.title(f"A = {A}")
x = np.linspace(-3,3)
y = x*3
plt.plot(x,y,color="black",alpha=0.3)
plt.legend([f"v = {v}",f"Av = {Av}","y=3x"])
plt.show()

In the above plots, Av (transformed vector) lies on the line \(y=3x\), and does not depend on the coordinates of vector v. As a result, using a low-rank matrix to transform a vector results only in the subspace of the original vector space. Here the subspace is the line \(y=3x\) and the original vector space is \(R^2\).