A repeated eigenvalue gives a continuum of Grassmann minimizers

For $q=\pm 1\in \mathrm{O}(1)$, $(xq{)}^{\mathsf{T}}A(xq)=x^{\mathsf{T}}Ax$, so the objective is well-defined on lines by [F1]. Since $A$ is positive semidefinite, $f([x])\ge 0$ for every unit $x$. If $x\in \mathrm{span}(e_1,e_2)$, then $Ax=0$ and $f([x])=0$, so every such line is a global minimizer.

The vectors $x=(\cos \theta )e_1+(\sin \theta )e_2$ for $0\le \theta <\pi$ are unit and give distinct lines: two such vectors span the same line only when their angles differ by a multiple of $\pi$, which cannot occur for distinct parameters in this half-open interval. Thus they parametrize continuum many distinct lines. At each such $x$, $Ax=0$, so the ambient gradient $2Ax$ and its horizontal projection vanish. The family is a continuum of Grassmann critical minimizers.