Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2015 / Paper 1 / 6

The spectral theorem for a real symmetric matrix gives real eigenvalues and an orthonormal eigenbasis. Compute the roots of $\det (A-\lambda I)=0$, find a basis for each eigenspace, and orthonormalize within any repeated-eigenvalue eigenspace. Eigenvectors belonging to distinct eigenvalues are already orthogonal: if $Av=\lambda v$ and $Aw=\mu w$, symmetry gives $(\lambda -\mu )v^{T}w=0$.

Putting the normalized basis vectors in the columns of an orthogonal matrix $Q$ yieldsFor repeated eigenvalues, the choice of axes inside their eigenspace is not unique.

The quadratic-form matrix has eigenvectorswith eigenvalues $1,9,9$. In the principal axes coordinates $\xi =(x_1+x_2)/\sqrt{2}$, $\eta =(x_1-x_2)/\sqrt{2}$ and $\zeta =x_3$, the equation becomesIt is a prolate spheroid, centred at the origin, with semiaxis three along $e_{+}$ and one in every perpendicular direction. The displayed $e_{-},e_3$ are one choice of short principal axes.

The sphere has equation ${\xi }^2+{\eta }^2+{\zeta }^2=4$. Combining it with the ellipsoid equation givesThe intersection of a prolate spheroid and a concentric sphere therefore consists of two circles in planes perpendicular to $e_{+}$. Each has radius $\boxed{\sqrt{5/8}=\sqrt{10}/4}$, and their centres areTheir plane equations in the original coordinates are $x_1+x_2=\pm 3\sqrt{3}/2$.

Minimize or maximize $D(x)=x^{T}x$ subject to $x^{T}Ax=1$. A Lagrange multiplier $\Lambda$ givesThe constraint excludes $x=0$ and implies $Ax\ne 0$, so it is a regular constraint surface. Multiplying the stationarity equation by $x^T$ gives $D=\Lambda >0$. Thus the distance extrema on a real quadratic surface satisfyEvery constrained extremum lies in a positive-eigenvalue eigenspace, with squared distance $D=1/\lambda$. If $A$ is positive definite, the global minimum is $1/{\lambda }_{\max }$ and the global maximum is $1/{\lambda }_{\min }$. For an indefinite or singular form the surface need not be bounded; eigenvector directions give stationary candidates, which may be saddles and need not supply both global extrema. With a repeated positive eigenvalue, there can be an entire family of equally distant stationary points.