Every Map Is a Rotation, a Stretch and a Rotation
Last timeDirections That Survive
Every matrix without exception turns out to be a rotation followed by stretching along perpendicular axes followed by another rotation, and the stretches are the numbers that matter.
The last lesson found directions a map leaves alone, and had to admit that some
maps do not have enough of them. This lesson gives up on asking for directions that
survive and asks a different question instead, and the answer has no exceptions at
all.
Ask which input direction the map stretches most. There is always one, since the
amount of stretching varies continuously over directions and directions form a
closed set. Then ask which direction perpendicular to that one is stretched most,
and continue. What falls out is a set of perpendicular input directions, each with
its own stretch factor, whose images are also perpendicular.
That is the whole result. Rotate the input so those special directions line up
with the axes, stretch each axis by its own factor, then rotate the result so the
stretched axes line up with where their images actually belong.
- the first rotation, which lines the special input directions up with the axes
- the stretch, one non-negative factor per axis and nothing off the diagonal
- the second rotation, which sends the stretched axes to where their images belong
- the original matrix, reproduced exactly
The numbers
The stretch factors are the content. There is one per direction, there are as many
as the smaller of the two dimensions, they are never negative, and they are
conventionally listed from largest to smallest.
The largest answers how much the map can magnify: no vector comes out more than
that many times longer than it went in, and some vector achieves it. The smallest
answers how much the map can shrink. Any factor that is exactly zero marks a
destroyed direction, so counting the non-zero factors gives the rank, which is the
first useful thing this list delivers.
| Map | Largest stretch | Smallest stretch | Rank | Trustworthy to invert |
|---|---|---|---|---|
| A rotation | 1 | 1 | full | yes |
| Flatten onto a line | 1 | 0 | one short | no |
| A tiny smallest stretch | 1 | 0.000001 | full | no |
| A table of measurements | large | nearly zero | full in principle | only the leading part |
A rotation has every factor equal to one, which is the formal way of saying it
changes no lengths. Flattening has one factor zero. The third row is the dangerous
case: every factor is non-zero, so the rank is full and the map is invertible, and
the smallest factor is small enough that the inverse is useless. The last row is
what measured data looks like, a few large factors and a long tail.
The lesson stops here
3 more paragraphs to go
You have read the opening. The rest of the argument, the problems that check whether it landed, and the lines worth keeping at the end all come with a plan.
The first lesson of every course in the library reads the whole way through, free, so you can see exactly what the rest of them are.
See the planThe contentsThis is the reading half
Starting the course gives you your own copy of it. Every idea on every page has problems standing under it, marked with a reason rather than a tick, and any sentence you do not believe can be opened and argued with. None of that can happen on a page nobody owns.
The contents