Partial Solutions to Brainfood #11

1. (a-lambda)(d-lambda) - bc
2. lambda^2 - trace lambda + det
3. Real if the discriminant trace^2 - 4 det >0.
4. For exactly one eigenvalue to be real and negative, we require the above condition and also
   trace - sqrt( trace^2 - 4 det ) < 0
   or
   det<0.
5. For real and both positive, we require condition (3) and
   trace - sqrt( trace^2 - 4 det ) > 0
   which is true for all trace > 0.
6. Purely imaginary for trace^2 - 4 det = 0.
7. Complex when trace^2 - 4 det < 0. The real part is exactly trace/2, so the real part is positive/negative as the trace is.
8. The eigenvalues of these matrices are degenerate or "in transition." The transitions include changing from being real to complex and changing from having positive real parts to negative real parts.