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title: Transformation Matrix
author: Erik-Jan van Kesteren
date: '2018-03-03'
slug: transformation-matrix
draft: false
- Web
- Matrix
- Web
- Matrix
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Any set of points can be represented in a matrix $\boldsymbol{X}$. For example:
$$ \boldsymbol{X} = \begin{bmatrix} 0 & 0 \\ 0 & 1 \\ 1 & 1 \\ 1 & 0 \end{bmatrix}$$
The four rows in this matrix correspond to four points in two-dimensional space. You can think of the first column as the x coordinate and the second column as the y coordinate of each point. For our chosen $\boldsymbol{X}$, these points represent the corners of a unit square.
We can define a transformation matrix $\boldsymbol{T}$ as a $2\times 2$ matrix which through post-multiplication transforms these points into _another_ set of points in 2-dimensional space $\boldsymbol{X'}$. For example, we can take the identity matrix:
$$\boldsymbol{T} = \boldsymbol{I} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$
This matrix is a kind of arbitrary transformation because by definition, $\boldsymbol{X'} = \boldsymbol{X} \times \boldsymbol{I} = \boldsymbol{X}$: the set of transformed points is the same as the set of original points.
But what about a different transformation matrix, say
$$\boldsymbol{T} = \begin{bmatrix} 1 & 0.5 \\ 0 & 1 \end{bmatrix}$$
Now $\boldsymbol{X'}$ is not equal to $\boldsymbol{X}$: the points have been transformed! In particular, here we are dealing with a _skew_:
$$\boldsymbol{X'} = \boldsymbol{X} \times \boldsymbol{T} = \begin{bmatrix} 0 & 0 \\ 0 & 1 \\ 1 & 1 \\ 1 & 0 \end{bmatrix} \times \begin{bmatrix} 1 & 0 \\ 0.5 & 1 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0.5 & 1 \\ 1.5 & 1 \\ 1 & 0 \end{bmatrix}$$
## Let's visualise it
Because this is all very abstract and a lot of numbers, below I've plotted the four points in $\boldsymbol{X}$, connected them by lines, coloured the center, and applied the _skew_ transformation, yielding $\boldsymbol{X'}$.
I've also gone a bit further and made it interactive[^1]. So you can edit the numbers in the matrix and the unit square will transform accordingly. Play around with it to get an idea of transforming a set of points in 2-dimensional space.
[^1]: Using JavaScript, SVG, and CSS. See the [source]( of this blog post for more info.
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## Determinant
Now that you have gained a feeling or intuition around the transformation matrix, I'll tell you a great geometric trick I learnt from [this youtube video]( the surface area of $\boldsymbol{X'}$ is equal to the size of the _determinant_ of the transformation matrix $\boldsymbol{T}$. This was a great revelation for me that made determinants much more easy to comprehend. This works in higher dimensions too: the transformed volume of a $k$-dimensional unit hypercube represents the size of the determinant of the transformation matrix $\boldsymbol{T} \in \mathbb{R}^k$.
But we're not there yet: determinants can be negative, wheras volumes and areas can't. Luckily, the sign of the determinant can be inferred from $\boldsymbol{X'}$ too. Specifically, it has to do with the _chirality_ of the shape defined by $\boldsymbol{X'}$. If the original square "flips" -- that is, the original bottom right point becomes the new top left point or the original bottom left point becomes the new top right point -- the sign of the determinant will be negative. In the illustration, that will make the shaded area red instead of blue.
The determinant of the currently entered $\boldsymbol{T}$ is <span id="det">0</span>.
## Top tips to try
1. Try to make $\boldsymbol{T}$ look like a covariance matrix.
2. Try to make the columns in $\boldsymbol{T}$ linearly dependent.
3. Try to flip the rows or columns of $\boldsymbol{T}$ at any point.
## Conclusion
Through exploring interactively what a transformation matrix does to a unit square, we can generate an intuition for the geometric meaning of the determinant.
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