I've been catching up on sleep over the weekend. I didn't get out of bed all of Saturday!
But my mind will not be put down so easily. Getting away from computer screens while the power is out has thrown me into full mathematical mode. Maybe this is some Freudian-level way of my subconscious craving more pen and paper work to satisfy some latent math craving. I even dreamed about a math problem!
Now, I've done that before. Most notably, I was staying up quite late to get a take home final for classical mechanics. I was concentrating so hard as I shut my eyes, that I woke up with the answer! I did this one other time more recently with a quantum fields homework. If only I could wake up with my thesis or that paper Kieron keeps asking me for!
I've also had dreams that posed math problems. For example, a trio of lighthouses whose beacons are spinning at different rates...what area is illuminated by the triangles bounded between the lighthouses and where they must meet if they're in an equilateral triangle? It took a few minutes after waking up to get that one.
However, this puzzle was more interesting! I was talking to someone who had one of those "evil" goatees and he wondered if I could find the eigenvalues of a matrix without doing the whole diagonalization procedure (i.e. avoid finding the characteristic polynomial). Challenged accepted!
I immediately leapt for the 2x2 case and actually solved it while I was still asleep. Taking the trace (sum of the diagonals) and the determinant (there's only one!) gives two coupled equations that can be solved for the eigenvalues (remember, the trace and determinant don't change under a similarity transform, so they're the same even when it's a diagonal matrix with the eigenvalues in the diagonal).
But evil goatee man wanted the 3x3 case!
This I actually had to wake up to check my answer. Honest, I was thinking about the inverse matrix in the dream, but numbers and writing don't work the same in a dream. So, it gets hard fast to legibly read anything (or even flip on a light switch!).
Since we now have three values to solve for, we need one more equation. The trace of the inverse matrix looks like the sum of the reciprocal eigenvalues, so voi'la! We win!
Awake me: 1, Asleep me: 0!
Now, you could quibble with me since my choice solution for solving for the inverse of a matrix is to find the characteristic polynomial and use the Cayley-Hamilton equation to obtain it (see this post), but there are other ways to find the inverse matrix...like seeing what row operations need to be performed to change it to the identity. I'm ok with this solution, and I guess I'll see how I was graded tonight.