Ask Uncle Colin: How do I look at matrices

Ask Uncle Colin is a chance to ask your burning, possibly embarrassing, maths questions — and to show off your skills at coming up with clever acronyms. Send your questions to and Uncle Colin will do what he can.

Dear Uncle Colin

How can you look at a matrix and tell what it does?

I just can’t see it, and I’m worried I’m going blind!

— Goddamned Opticians, Dreadful Eyesight Loss

Since GODEL is studying FP1, I’ll limit this discussion to 2D matrices, although the same sort of idea works for 3D (or, for that matter, nD) matrices.

Hi, GODEL — don’t worry, studying matrices won’t make you go blind (although I’d suggest making sure you do it in brightly-lit places just in case).

As for how to visualise what’s going on, if you were to give the Mathematical Ninja a matrix — let’s say, $\left( \begin{matrix} \frac 12 & – \frac 12 \sqrt{3} \\ \frac 12 \sqrt{3} & \frac12 \end{matrix} \right)$ — he would immediately tell you it was an anticlockwise rotation of $\frac 13 \pi$ about the origin. How?

The trick is to see what happens when you apply the matrix to two specific vectors: $\colvectwo{1}{0}$ (the x-axis, if you like), and $\colvectwo{0}{1}$ (the y-axis). As it turns out, the first of those is the first column of the matrix; the second is the second column.

So, when the Mathematical Ninja sees $\mattwotwo{\frac 12&}{-\frac 12\sqrt{3}}{\frac12 \sqrt{3}&}{\frac 12}$, he notes that the first column is $\colvectwo{\frac 12}{\frac 12 \sqrt 3}$, which could be a rotation of $\frac 13 \pi$ around the origin, a reflection in the line $\sqrt 3 y – x = 0$, or a large number of other possibilities.

Checking the second column shows that the point on the y-axis moves to $\colvectwo{-\frac12 \sqrt 3}{\frac 12}$, which fits with the rotation hypothesis, but not the others!

So, rather than remembering the rather unmemorable formulas for reflection and rotation matrices, you can simply look at the matrix, maybe draw a little sketch, and see what’s going on!

— Uncle Colin


Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.


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