A field guide to matrices
Birders don't learn birds from definitions. They learn the common ones by sight, then the field marks that tell look-alikes apart. You can learn matrices the same way.
Every $2 \times 2$ matrix is a way of moving the plane, and there are only a handful of species. Each plate below shows one of them in action. Click a plate to watch it happen.
where the grid lines go
where the unit circle goes
where $e_1$ goes
eigenvector lines
the F shows whether the plane got flipped
What the labels on each card mean
- det
- The determinant: how much the matrix scales areas. 2 means areas double, 0 means everything is squashed flat, and a minus sign means the plane gets flipped over (watch the F).
- trace
- The two diagonal entries added together. It always equals the sum of the eigenvalues.
- eigenvalues
- How much the matrix stretches along its eigen lines. A negative one flips that direction. Complex ones (with i) mean no direction survives without being turned.
- eigen lines
- Lines through the centre that the matrix maps onto themselves: it only stretches along them. They are the dashed yellow lines on the plate.
- field marks
- What to look for to recognise it.
- habitat
- Where it turns up in maths and in the world.
- mistaken for
- The look-alike it gets confused with, and how to tell them apart.
Identify a specimen
Type in any $2\times2$ matrix and the key walks it down to its species, one yes-or-no question at a time.
M =
try: