Partial Solutions to Brainfood #11
These are partial answers so that you can check to see if you are on target.
We expect your actual answers to be "fleshed out" a bit more.
- (a-lambda)(d-lambda) - bc
- lambda^2 - trace lambda + det
- Real if the discriminant trace^2 - 4 det >0.
- For exactly one eigenvalue to be real and negative, we require the
above condition and also
trace - sqrt( trace^2 - 4 det ) < 0
- For real and both positive, we require condition (3) and
trace - sqrt( trace^2 - 4 det ) > 0
which is true for all trace > 0.
- Purely imaginary for trace^2 - 4 det = 0.
- 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.
- 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.
Last modified: Fri Nov 22 11:06:06 1996