classes were good today; in analysis we started doing all the calculus stuff, differentiation and mean value theorem, multivariable calculus and taylor's theorem etc. in linear algebra we started talking about diagonalization of n by n matrices. there's a cool proof that a real symmetric n by n matrix (equivalently, any self-adjoint linear operator on a vector space over the reals) has a real eigenvalue. (actually, in class we proved something stronger, but i liked this first part of the proof best.) i am going to try to catch up on some linear algebra stuff tonight by typing up notes, and i'm going to read some of the calculus stuff on my own because i thought the presentation was a little rushed and a bit unclear in class...
i'm currently doing laundry since i'm running out of socks. i'm also making some french fries! potatoes are currently soaking... that plus a spinach salad should make for a decent dinner. i am really running low on food though; i'm just going to have to make a legitimate trip to the store someday soon. cardinals are in town tonight; if i felt like it, i bet i could find a TV to watch it on but i will probably just listen. anyway, all is well! thanks for reading! by the way, if you're interested in how the proof i mentioned works, i'd be glad to discuss it. originally i posted up a proof, but there were formatting mistakes because of typing in html, and so i did a proof in latex but then i made a few mistakes, and didn't save the file, etc. etc. etc.; so i decided to forget it!
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What's cool about it?
ReplyDeleteI wish I could take you to the grocery store! Maybe they deliver!
ReplyDeletei think it's cool because you basically use analysis results to prove it. you define F(v)=(Tv,v) where T is the matrix/linear transformation; and this F is a continuous function, and you define it only on the unit sphere, so that you have a continuous function on a compact set. then analysis tells you that that means F attains a maximum at some v_0. then you basically pick w with (w,v_0)=0 and define a function g(t) that depends on v_0 and w such that g(0)=v_0 and g(t) is in the unit sphere for each t. g is differentiable, and so is F, so F(g) is differentiable, and since it attains a maximum at t=0, its derivative at t=0 is 0... then when you actually do that differentiation you find out that (Tv,w)=0. so if w is perpendicular to v, then Tv is perpendicular to w. so basically, intuitively, if Tv is perpendicular to anything that v is perpendicular to, then they must be pointing in the same direction. which means Tv=cv for some real number c, which means v is an eigenvector with real eigenvalue c.
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