Past exam of the mathematics course of the University of Cambridge / 2026 / ib / Paper 1 / 8E (Linear Algebra)

A Jordan form is block diagonal with blocks $J_k(\lambda )$ having $\lambda$ on the diagonal and ones on the superdiagonal. Over $\mathbb{C}$ every square matrix is similar to one, uniquely up to permutation of blocks.

$D$ lowers total degree, so it is nilpotent and has only eigenvalue zero. Solving $({\partial }_z+{\partial }_w)f=0$ gives $f=h(z-w)$; the degree restrictions give the eigenspace $\mathrm{span}\{1,z-w,(z-w{)}^2\}$.

Direct degree counting gives $\dim \ker D^k=3,5,7,8,9$ for $k=1,\dots ,5$. Hence the Jordan block sizes are $5,3,1$.

On each Jordan block of $A$, $g(A)$ is triangular with diagonal entry $g(\lambda )$; hence the eigenvalues, with multiplicity, are $g({\lambda }_i)$.

The shift matrix obeys $A^4=I$ and $B=aI+bA+c{A}^2+d{A}^3$. Evaluating $g(t)=a+bt+c{t}^2+d{t}^3$ at $1,i,-1,-i$ gives