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wu :: forums    riddles    putnam exam (pure math) (Moderators: Grimbal, SMQ, Icarus, towr, william wu, Eigenray)    Constant Norm « Previous topic | Next topic » Pages: 1  Reply Notify of replies Send Topic Print    Author  Topic: Constant Norm  (Read 482 times) william wu wu::riddles Administrator     Gender: Posts: 1291 Constant Norm   « on: Feb 8th, 2006, 8:44pm » Quote Modify Consider the matrix differential equation x' = A x where x is a state vector in Rn, x' is the derivative of that vector with respect to time, and A is an n-by-n matrix. Find conditions on A which guarantee that the Euclidean norm || x ||2 remains constant.   Source: Stephen Boyd, Stanford EE Quals 2006 « Last Edit: Feb 9th, 2006, 1:13am by william wu » IP Logged [ wu ] : http://wuriddles.com / http://forums.wuriddles.com Eigenray wu::riddles Moderator Uberpuzzler     Gender: Posts: 1948 Re: Constant Norm   « Reply #1 on: Feb 8th, 2006, 9:50pm » Quote Modify o(n)?  (That is, the Lie algebra of O(n), i.e., skew-symmetric matrices.) « Last Edit: Feb 8th, 2006, 9:54pm by Eigenray » IP Logged towr wu::riddles Moderator Uberpuzzler Some people are average, some are just mean.     Gender: Posts: 13730 Re: Constant Norm   « Reply #2 on: Feb 9th, 2006, 12:50am » Quote Modify Which norm is that? ||x||2 (i.e euclidean distance?) IP Logged Wikipedia, Google, Mathworld, Integer sequence DB william wu wu::riddles Administrator     Gender: Posts: 1291 Re: Constant Norm   « Reply #3 on: Feb 9th, 2006, 1:21am » Quote Modify Yes sorry, Euclidean norm.     That's right Eigenray. I thought it was a cute result IP Logged [ wu ] : http://wuriddles.com / http://forums.wuriddles.com Eigenray wu::riddles Moderator Uberpuzzler     Gender: Posts: 1948 Re: Constant Norm   « Reply #4 on: Feb 9th, 2006, 9:14am » Quote Modify To elaborate, we have x(s) = esAx, so we want esA to be orthonormal for all s, which is precisely the statement that A is in the Lie algebra o(n) associated to the Lie group O(n).   esA is orthonormal when I = (esA)t esA = esA^t esA. Differentiating wrt s, and since At commutes with eA^t, we have esA^t (At+A) esA = 0, and so we find At+A = 0.  For sufficiency, run the argument backwards, or just note that eA is clearly orthonormal when At = -A: since A and -A commute, (esA)t esA = e-sAesA = eO = I.   By the same argument, if we had x in Cn, we'd want A to be in u(n), the Lie algebra of U(n), i.e., skew-Hermitian. IP Logged william wu wu::riddles Administrator     Gender: Posts: 1291 Re: Constant Norm   « Reply #5 on: Feb 10th, 2006, 2:48am » Quote Modify Another way:   - Want || x || to be constant, so, the derivative of ||x|| with respect to time should be zero. That is,   0 = (d/dt) [ || x || ] = (d/dt) [ sqrt( xT x ) ]     - Just throw away the square root. Thus we need     0 = (d/dt) [ xT x ] = 2 xT x' = 2 xT A x   So 0 = xT A x.       - Recall  xT A x = xT ((A + AT)/2) x. So when A = -AT, the quadratic form is zero. IP Logged [ wu ] : http://wuriddles.com / http://forums.wuriddles.com Pages: 1  Reply Notify of replies Send Topic Print Forum Jump: ----------------------------- riddles -----------------------------  - easy   - medium   - hard   - what am i   - what happened   - microsoft   - cs => putnam exam (pure math)   - suggestions, help, and FAQ   - general problem-solving / chatting / whatever ----------------------------- general -----------------------------  - guestbook   - truth   - complex analysis   - wanted   - psychology   - chinese « Previous topic | Next topic »

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