= Solution
{c}
Interpret diagonalizability over the specified underlying field $\mathbb R$. Write $A=P\operatorname{diag}(\lambda_1,\ldots,\lambda_n)P^{-1}$ with real eigenvalues. Invariance of the <trace> under <similarity> gives
$$
\varphi(A,A)=\operatorname{tr}(A^2)=\sum_i\lambda_i^2.
$$
If $A\ne0$ is diagonalizable, at least one eigenvalue is nonzero, proving \b[$\varphi(A,A)>0$]. 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.
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