Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2011 / Paper 1 / 13

For the two-by-two matrix, take the product of the diagonal entries minus the product of the off-diagonal entries:In particular, $A$ is an invertible matrix.

Interchange the diagonal entries, negate the off-diagonal entries, and divide by the nonzero determinant. The matrix inverse isMultiplication by $A$ gives the identity matrix, since each diagonal product sum is $25-9=16$ and each off-diagonal product sum is zero.

The characteristic polynomial isFor $\lambda =2$, the eigenvector equation reduces to $x=y$; for $\lambda =8$, it reduces to $x=-y$. Choose the following unit vectors consistently for the remaining parts:Each eigenspace consists of all scalar multiples of its displayed vector, including zero; an eigenvector itself must be nonzero.

The chosen eigenvectors have dot productThey are therefore orthogonal, and their unit lengths make them an orthonormal eigenbasis. This agrees with the spectral theorem for real symmetric matrices, since $A$ is real symmetric and its two eigenvalues are distinct.

Multiply the quadratic form out directly:Thus $b=5$, $c=-6$ and $d=5$. Both eigenvalues are positive, so this is a positive-definite quadratic form; its positive unit level set is an ellipse. The next part exhibits its principal axes explicitly.

Let $Q=(x_1{x}_2)$ for the normalized eigenvectors already chosen. Then $Q^{T}Q=I$, $(x,y{)}^T=Q(x^{\prime },y^{\prime }{)}^T$, and orthogonal diagonalization givesThe ellipse has semi-axis lengths $1/\sqrt{2}$ and $1/\sqrt{8}$ along $x_1$ and $x_2$, respectively. Rescale $x^{\prime \prime }=\sqrt{2}{x}^{\prime }$ and $y^{\prime \prime }=\sqrt{8}{y}^{\prime }$ to obtain the unit circle $(x^{\prime \prime }{)}^2+(y^{\prime \prime }{)}^2=1$. Since $(x^{\prime },y^{\prime }{)}^T=Q^T(x,y{)}^T$,The identity $B^{T}B=A$ checks the rescaling directly. Swapping or reversing eigenvectors gives other valid coordinate matrices; the displayed one follows the explicit choices in part (a).