Tied PCA eigenvalues and sign ambiguity

The eigenvalues of ${\mathrm{\Sigma }}_1=I_2$ are $1,1$, and $I_2-I_2$ is the zero matrix, so the defined leading eigenspace is $E_1=\ker (I_2-I_2)={\mathbb{R}}^2$. For any finite-second-moment random vector $X$ with covariance ${\mathrm{\Sigma }}_1$ and every unit vector $a\in {\mathbb{R}}^2$, [F4] gives $\mathrm{Var}(a^{\mathsf{T}}X)=a^{\mathsf{T}}{I}_{2}a=\mathrm{\parallel }a{\mathrm{\parallel }}_2^2=1$. Thus every unit vector in $E_1$ attains the same projected variance $1$ and maximizes it; each of its lines is a variance-maximizing direction, while the defined leading PCA subspace remains all of ${\mathbb{R}}^2$.

The eigenvalues of ${\mathrm{\Sigma }}_2=\mathrm{diag}(2,1)$ are $2>1$, with leading eigenspace $\mathrm{span}(e_1)$. Both $e_1$ and $-e_1$ have unit norm and satisfy ${\mathrm{\Sigma }}_2{e}_1=2e_1$ and ${\mathrm{\Sigma }}_2(-e_1)=2(-e_1)$. They determine the same line, so even a simple leading eigenvalue does not orient its eigenvector.