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Mathematical methods I course of the University of Oxford

Past exam of the mathematical methods I course of the University of Oxford

2014

Put $p=\epsilon+\delta+\eta$ and $q=\epsilon\delta+\delta\eta+\eta\epsilon$. Direct expansion gives the <characteristic polynomial>
$$
\det(\lambda I-M)=(\lambda+\epsilon)(\lambda+\eta)(\lambda+\delta)-\epsilon\eta\delta
=\lambda(\lambda^2+p\lambda+q).
$$
The quadratic formula therefore gives
$$
\boxed{K=-\frac{\epsilon+\delta+\eta}{2},\qquad W=\epsilon\delta+\delta\eta+\eta\epsilon,\qquad\lambda_\pm=K\pm\sqrt{K^2-W}.}
$$
Since $p,q>0$, both nonzero <eigenvalues> have negative <real part>. They need not be real; equal rates give a complex-conjugate pair.

Solution

Past exam of the mathematical methods I course of the University of Oxford / 2014 / 11 / ii / Solution

Past exam of the mathematical methods I course of the University of Oxford / 2014 / 11 / ii / Solution

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Past exam of the mathematical methods I course of the University of Oxford / 2014 / 11 / ii / Solution

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