Past exam of the mathematics course of the University of Cambridge / 2010 / ia / Paper 1 / 5A / ii

The matrix trace calculation in (i) givesA sum of nonnegative real numbers is zero exactly when each is zero. Thus $\boxed{\mathrm{tr}(A^{T}A)=0⟺A=0}$, so the real Frobenius inner product is positive-definite.

Write $\alpha =\mathrm{tr}(A^{T}A)$, $\beta =\mathrm{tr}(B^{T}B)$ and $\gamma =\mathrm{tr}(A^{T}B)$. If $B\ne 0$, then $\beta >0$, and applying the same nonnegativity to $A-tB$ givesTaking $t=\gamma /\beta$ yields $\alpha -{\gamma }^2/\beta \ge 0$. If $B=0$, the claimed inequality is immediate. Hence the Cauchy-Schwarz inequality here isWhen $B\ne 0$, equality holds exactly when $\mathrm{tr}((A-(\gamma /\beta )B{)}^T(A-(\gamma /\beta )B))=0$, hence exactly when $A=(\gamma /\beta )B$. Together with the case $B=0$, the complete equality condition is $\boxed{A,B\text{ are linearly dependent}}$, including the case that either is zero.