Past exam of the algebra, analysis and differential equations 1 course of the University of Oxford / 2010 / 2 / d / Solution

Interpret diagonalizability over the specified underlying field $\mathbb{R}$. Write $A=P\mathrm{diag}({\lambda }_1,\dots ,{\lambda }_n)P^{-1}$ with real eigenvalues. Invariance of the trace under similarity givesIf $A\ne 0$ is diagonalizable, at least one eigenvalue is nonzero, proving $\phi (A,A)>0$. Real diagonalizability is essential: the rotation block from part (c) is diagonalizable over $\mathbb{C}$ but has eigenvalues $i,-i$ and negative trace of its square.