Damped Harmonic Oscillator [ Math not physics question] Question
Consider, The motion of a a damped harmonic oscillator is described by $x^{''}$ + $bx^{'}$ + $kx = 0$, where b is greater than or equal to 0
A) Rewrite as a two dimensional linear system.
$ x^{'} = y$
$y^{'}$ = - $(by + kx)$
B) Sketch the analogue of the trace-determinant plane in the bk-plane. That is, identify
the regions in the relevant portion of the bk-plane where the corresponding system has similar phase portraits.
I have cut out most of my answer etc as it seems to of merely confused people, how does one answer this question?

For A, your rewrite as a two dimensional system is fine.  I don't understand where the vectors come from and although you say "eigenvalues are given by" you have only one expression.  
For B, I don't know what a Trace-Determinate Parabola picture is.
For C, it depends on the roots of the quadratic $mz^2+bz+k=0$  If the roots are complex, it is underdamped as the exponential of the imaginary part gives oscillation.  It is critically damped if the roots are equal and overdamped if they are real and different.
