Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2016 / Paper 1 / 14 / b

The third coordinate is decoupled, giving the unit eigenvector $(0,0,1{)}^{\mathsf{T}}$ with eigenvalue $1$. In the first two coordinates, the symmetric direction $(1,1)$ has eigenvalue $3-1=2$, and the antisymmetric direction $(1,-1)$ has eigenvalue $3+1=4$. Their normalized forms areThey are an orthonormal basis of ${\mathbb{R}}^3$; independent sign reversals are equally valid choices.

Insert the eigenvalues and eigenvectors into the inverse spectral decomposition. The first two projections contributeAdding the third projection givesMultiplication by $A$ gives the identity; in particular the upper block satisfies $\left(\begin{array}{cc}3& -1\\ -1& 3\end{array}\right)\left(\begin{array}{cc}3& 1\\ 1& 3\end{array}\right)=8I$.

Summing the diagonal entries givesThe eigenvalues of the inverse are the reciprocals $1/2,1/4,1$ of those of $A$, whose sum is also $7/4$. This is precisely what the trace identity predicts.