before and after applying a linear transform to a space.
each of those vectors has been stretched in the process.
and then we're going to try and extend this concept into three dimensions.
which is where we scale by the same amount in each direction.
any vector would be an eigenvector.
we're going to look at rotation.
and we found that it had no eigenvectors.
and that is 180 degrees.
but just pointing in the opposite direction.
they are all now pointing in the opposite direction.
and it's slightly less obvious than some of the previous examples.
the green horizontal vector is an eigenvector and its eigenvalue is still one.
this transformation does have two eigenvectors.
eigenvectors aren't always easy to spot.
watch our parallelogram go back to its original square.
But this time, with our eigenvector visible.
an eigenvector as it stays on its own span.
the system as being composed of hundreds of dimensions or more.
a more robust mathematical description of this concept to allow us to proceed.
Before we do, let's just take a look at one quick example in 3D.
operate in much the same in 3D as they do in 2D.
However, rotation does take on a neat new meaning.
the orange vector has not moved.
it means we've also found the axis of rotation.
going to go about writing a formal definition of an eigen-problem.
And this is exactly what we're going to be discussing next time. See you then.