CVS difference for ai05s/ai05-0047-1.txt

--- 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).
+
 ****************************************************************