Matrix Transformation

Nov 01, 2021 permanent    LinearAlgebra

We can think of a matrix as a transformation of a Vector or all vectors in space.

When we take the product of a matrix and a vector, we are transforming the vector.

A transformation is another word for a function: it takes in some inputs (a vector) and returns some output (a transformed vector).

For example, we can rotate a vector [xy]\begin{bmatrix}x \\ y\end{bmatrix} some angle θ\theta about the origin using a Rotational Matrix.

[x*y*]=[cos⁡θsin⁡θ−sin⁡θcos⁡θ][xy]\begin{bmatrix}\text{x*} \\ \text{y*} \end{bmatrix} = \begin{bmatrix}\cos\theta && \sin\theta \\ -\sin\theta && \cos\theta\end{bmatrix} \begin{bmatrix}x \\ y\end{bmatrix}

In this example, we perform a 90° rotation of vector [22]\begin{bmatrix}2 \\ 2\end{bmatrix}.

Rotation matrix product with vector

To describe a transformation as a matrix, we only need to record where the Basis Vectors land as columns of a new matrix: [abcd]\begin{bmatrix}\color{red}{a} && \color{green}{b} \\ \color{red}{c} && \color{green}{d}\end{bmatrix}

For example, a Shear Transformation keeps the i^\hat{i} basis vector fixed, and slants the j^\hat{j} basis vector. We can record that as: [1102]\begin{bmatrix}\color{red}{1} && \color{green}{1} \\ \color{red}{0} && \color{green}{2}\end{bmatrix}

Transformed basis vectors

A matrix transformation is always linear in that it keeps all gridlines in space are parallel and evenly spaced.

[@3blue1brownVectorsChapter3Essence2016]

[@dyeMathematicsMachineLearning]

Image processing is a use case for matrix transformations.

Since we represent an image as a m x nm \ x \ n grid of pixels, we can treat the position of each pixel as a vector, then perform a transform of each position vector to transform the entire image.

In this example, I rotate an image using the rotational matrix above.

There's some additional code required:

  • Create a new matrix that's the maximum possible width and height.
  • Convert each position into a vector.
  • Convert each position vector, so it's a distance from the center, not top-left.
  • Rotate each position.
  • Revert convert using a new image size.

Note that we end up with some empty pixels in a 45° rotation. These occur because some of the transformed coordinates are floating-point numbers. When they get rounded into integer positions, some of the pixels get excluded. There are many strategies to deal with this, but that's for another article.

[@agrawalRotatingImage]

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