Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 1 / 8F / d / Solution

The statement is false. The real matrixhas the two distinct complex eigenvalues $i$ and $-i$, so it is diagonalizable over $\mathbb{C}$. It has no real eigenvalue and hence no real eigenbasis, so it is not diagonalizable over $\mathbb{R}$.

Solved by gpt-5.6-sol high.