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*:

\[\boldsymbol{X'} = \boldsymbol{X} \times \boldsymbol{T} = \begin{bmatrix} 0 & 0 \\ 0 & 1 \\ 1 & 1 \\ 1 & 0 \end{bmatrix} \times \begin{bmatrix} 1 & 0 \\ 0.5 & 1 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0.5 & 1 \\ 1.5 & 1 \\ 1 & 0 \end{bmatrix}\]

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.

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 0.

- Try to make \(\boldsymbol{T}\) look like a covariance matrix.
- Try to make the columns in \(\boldsymbol{T}\) linearly dependent.
- Try to flip the rows or columns of \(\boldsymbol{T}\) at any point.

Through exploring interactively what a transformation matrix does to a unit square, we can generate an intuition for the geometric meaning of the determinant.