Change of eigenvalues under near orthogonal matrix multiplication

Question

Let $A,B\in\mathbb{R}^{d\times d}$ be positive definite diagonal matrices. Let $\Phi\in\mathbb{R}^{n\times d}$ satisfy $\text{rank}(\Phi) = n \leq d$ and be such that $\Phi\Phi^T = I_n$. When $d = n$, $\Phi$ is orthonormal and hence $$\lambda_i(\Phi A\Phi^T) - \lambda_i(\Phi B\Phi^T) = \lambda_i(A) - \lambda_i(B),$$ where $\lambda_i(\cdot)$ denotes the $i^{\text{th}}$ largest eigenvalue. If $n<d$, is it possible to express (or bound) $$ \lambda_i(\Phi A\Phi^T) - \lambda_i(\Phi B\Phi^T)$$ in terms of $\lambda_i(A) - \lambda_i(B)$ and $d-n$ ?

Answer 1

I believe that if $$A=\left(\begin{matrix} a_1 & 0 \\ 0 & a_2 \end{matrix}\right)$$ where $a_1>a_2>0$, and $\Phi = (0\ 1)$, then $\lambda_1(A) = a_1$ and $$\lambda_1(\Phi A \Phi^T + \alpha) = \Phi A \Phi^T + \alpha = a_2 + \alpha.$$

So, $$\lambda_1(\Phi A \Phi^T + \alpha) - \lambda_1(A) = a_2 +\alpha - a_1$$ which cannot be bounded by a function of just $\lambda_1(A)=a_1, \alpha, d,$ and $n$.