Let be an real symmetric matrix. Let be an eigenvalue and be a corresponding eigenvector. Then
We deduce that . And is clearly a positive real, so .
This immediately shows that the characteristic polynomial of has only real roots.
It is also easy to see that eigenvectors for distinct eigenvalues are orthogonal. If and , then is both and , so and .
The remaining point is to rule out the possibility of nontrivial Jordan blocks or, in the language of intro linear algebra classes, the possibility that geometric multiplicity is less than algebraic multiplicity. For a class where the Jordan canonical form theorem is proved, this is simple enough. If there is a nontrivial Jordan block, then there is some such that but . But then
so , contradicting that .
However, the Linear Algebra course that I am currently teaching doesn’t do Jordan canonical form. I haven’t figured out how to show that geometric mult. less than algebraic mult. implies that simply enough to fit in a lecture where I also talk about the rest of this stuff.
Nonetheless, I am amazed that every textbook I have seen uses the "optimize a quadratic form on the unit ball" argument rather than this algebraic once. Lots of students don't remember multivariable calculus well, and existence of maxima of continuous functions on multidimensional bounded domains is complicated. Plus, I find a lot of students have trouble with an inductive process like getting one eigenvector and splitting off an orthogonal complement.
This argument is just shuffling algebra around, combined with the fact that a sum of squares is nonzero. It seems clearly easier to me.
I'm just a little too short on time to present the whole argument this term. I'll do the two easy parts above and hand wave at the algebraic versus geometric issue. Next time I teach this course, I'll try to plan my discussion of algebraic multiplicity better so that I get the key lemma in.
I'm excited! I had thought that the spectral theorem would just be way too hard to prove in this class, but I think this will at least do a large part of it.