Oh, thanks for the clarification.
Imagine a linear transformation in the 3-dimensional space; that means that rotations, "flips", and scalings are allowed. For example take a rotation: points that are in the axis of rotation remain on that line. Likewise, think of the plane orthogonal to the axis of rotation, then points that are on that plane rotate, but they stay on that plane. The axis of rotation and its orthogonal plane are both invariant subspaces.
In a 3-dimensional space, and in general in every odd-dimensional space, every linear transformation has an invariant subspace. This is not true for even-dimensional spaces, such as the 2d plane: a 45-degree rotation for example has no invariant subspace (except for {0} and the whole space, of course). However, if you allow the coordinates to be complex numbers, instead of real, then there always is an invariant subspace, although it is harder to visualize.
To express this in linear algebra terms, every complex matrix has an eigenvector. This is a fundamental fact that has applications pretty much everywhere in mathematics.
The Invariant Subspace problem generalizes this statement to spaces that are infinite-dimensional. Infinite-dimensional spaces are infinitely messy, so we restrict our attention to spaces that have some additional structure; for example Hilbert spaces, those addressed in the paper, are those where a formula very similar to the Pythagorean theorem is true.
I very much doubt that it makes much sense to anyone who does not know linear algebra reasonably well.
It's a lot to digest. As I explained in another reply, my mathematics is very poor indeed.
> Likewise, think of the plane orthogonal to the axis of rotation, then points that are on that plane rotate, but they stay on that plane.
Why only the orthogonal plane, I don't see how any set of points behaves differently to any other. For example the stars could be said to rotate around the earth (I know, but pretend). They don't move in relation one another though. They only change relative to the earth. I think I'm grossly misunderstanding. Does 'rotation' have a special meaning in this context?
> 45-degree rotation for example has no invariant subspace
because you can rotate points on a line 45 degrees and the only one that stays the same is 0,0, which you call {0}?
So.. someone tried to prove that for n dimenional (odd?) spaces with n<=infinity, there is always some subspace which doesn't differ when you apply complex transformations to it? But was wrong.
http://www.amazon.com/Concepts-Modern-Mathematics-Ian-Stewar...
Same thing is true of the axis of rotation.
All other pairs of points will be in a different plane than they were before the rotation.
Does that help?
I always feel like I'm missing detail, eg:
> every complex linear transformation
Is that "complex" important?
> line that gets mapped to itself
So a trivial example might be a 360 degree rotation? Or am I just way misunderstanding this stuff?
* and I do mean real CS, not software engineering.
I don't think my tutors at Oxford really understood exactly what I meant by "I can't do all this maths". Basic gaps in my knowledge. Still passed though, fuck yeah! :D
Because (-1)^2 = 1^2 = 1, these numbers don't exist on the regular ("real") number line. Instead we wave our hands and posit the existence of the magical number i, the imaginary unit, that has the property i^2 = -1. Through simple algebra, this unit is enough to give us square roots for all the imaginary numbers. For example, (3i)^2 = -9.
Imaginary numbers are "independent" of the real numbers, which means that the sum of 5 and 3i is simply 5+3i. That expression cannot be simplified further. Such sums are called "complex" numbers. The complex numbers can be imagined as a plane, with the horizontal axis consisting of the pure reals, and the vertical axis consisting of the pure imaginaries.
Complex numbers support all the regular arithmetic operations. You can figure them out yourself, so long as you remember i^2 = -1. In the complex plane, addition looks like vector addition, and multiplying by an imaginary number is a rotation.
>> line that gets mapped to itself
> So a trivial example might be a
> 360 degree rotation?
A rotation in 2D space apparently only leaves the origin fixed, and that's where we need complex numbers. To find a fixed line in that case we need to find roots of a quadratic with negative discriminant, hence needing sqrts of negative numbers.
Edit: clearly it's possible though, as the other replies attest to.
Several times I began trying to write an actual explanation of what a vector space is, a linear function, a subspace, a Banach space, a Hilbert space, and what the actual conjecture is. I got bogged down every time.
Besides, I read sarcasm, not snark.
PS: Now that's a snarky tone. My original comment was merely lightly sarcastic.