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I sense some sarcasm :) Luckily these things have nice geometrical interpretations, let me try to clarify this better, hoping that I can stimulate some curiosit
by ot 14y ago
I sense some sarcasm :) Luckily these things have nice geometrical
interpretations, let me try to clarify this better, hoping that I can
stimulate some curiosity on the subject.
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.
- btilly 14y agoI know all of the math you discussed, so for me what you said makes perfect sense. I very much doubt that it makes much sense to anyone who does not know linear algebra reasonably well.
- tsewlliw 14y agoI'm curious, but I think I'd need a practical example
- user24 14y agoThanks for your detailed reply. 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.
- nickpinkston 14y agoTo get better intuition without a math degree I'd recommend: Concepts of Modern Mathematics which only requires elementary algebra, but will give you great intro treatments of everything from groups to number theory. http://www.amazon.com/Concepts-Modern-Mathematics-Ian-Stewart/dp/0486284247/ref=pd_sim_b_2 http://www.amazon.com/Concepts-Modern-Mathematics-Ian-Stewar...
- neutronicus 14y agoThe important part about the orthogonal plane is that when you rotate it, all pairs of points that were in the orthogonal plane before are still in the orthogonal plane. 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.