--- ai05s/ai05-0047-1.txt 2007/08/03 04:14:09 1.3 +++ ai05s/ai05-0047-1.txt 2007/08/07 01:16:17 1.4 @@ -154,4 +154,73 @@ produces two identical floating-point values, then the whole thing is probably ill-conditioned, so you get what you get. +**************************************************************** + +From: John Barnes +Sent: Friday, August 3, 2007 2:17 AM + +Perhaps you would feel happier about eigenvalues with an example of their +use in nature. They turn up in all sorts of physical situations. Perhaps the +simplest is the moment of inertia of a solid object such as the earth. Now +the earth is a slightly flattened sphere. As a consequence its moment of +inertia about the polar axis is more than the moment of inertia about an +axis going through the equator. Moreover, it will spin smoothly about any of +these axes. But if you atttempt to spin it about an arbitrary axis like an +axis through Madison, Paris or London then it will wobble because it is not +symmetric about that axis. + +The axes where there is no wobble are known as the principal axes. The +moments of inertia about them are the eigenvalues and the axes themselves +are the eigenvectors. + +In the case of an oblate sphere like the earth, one principal axis is the +polar axis and any two at right angles through the equator can be taken as +the other axes. That's because two of the eigenvalues are the same. A +general rigid body will have three distinct eigenvalues and no confusion +about the axes. + +The eigenvalues are easy and there is never any dispute about them. The +problem is the eigenvectors because for one thing you can take them in +either direction. From North to South pole or vice versa for example. And if +two or more eigenvalues coincide then the axes can be chosen in lots of +different ways. + +These eigensystems turn up in lots of physical situations. One other that +immediately springs to mind is in elasticity. Something like a block of wood +has a different elasticity along the grain and across the grain. + +They also turn up in statistics when estimating several parameters. + +The complex number stuff turns up in quantum mechanics. + +Her endeth the first lesson. + +--- + +Another example I should have mentioned is an aeroplane propeller. If it has +just two blades then the three moments of inertia (three eigenvalues) are +different. I believe this causes vibration when turning. A four bladed +propellor has two eigenvalues the same and no vibration. A three bladed +propellor is OK as well perhaps surprisingly. + +--- + +> I for one cannot think of a practical application of complex eigenvalues in +> finite-dimensional spaces. + +Got it - I knew it was something about angular momentum. + +The Pauli spin matrices are Hermitian. See for example Penrose, The Road to +Reality page 550-551. The matrices are (using good old Ada aggregates) + +L1 = ((0, 1), (1, 0)) +L2 = ((0, -i), (i, 0)) +L3 = ((1, 0), (0, -1)) + +OK so L1 and L3 are only real, But L2 is the proper thing. It is about the +simplest posssible Hermitian matrix. + +The eigenvalues are the possible spin values, which are in fact real +(half-integer, really). + ****************************************************************

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