linear algebra – Formula for $mbox{trace}(X_2^{-2} X_1)$, given +ve definite matrices of the form $X_i = a_i 1_n 1_n^top + b_i A + c_i I_n$ with $a_i,b_i,c_i ge 0$

Let $A$ be an $n times n$ psd matrix and $a_1,b_1,c_1,a_2,b_2,c_2$ be nonnegative constants with $c_2>0$. Consider the matrices $X_i := a_i 1_n 1_n^top + b_i A + c_i I_n$ for $ i = {1,2}$, where $1_n1_n^top$ is the $n times n$ matrix with all entries equal to $1$.

Question. Is it possible to write $mbox{trace}(X_2^{-2} X_1)$ as a sum of ratios of eigenvalues of (powers of) matrices of the form $X_i$ ?

Example. Here are some examples of the kinds of results I’m interested in.

  • If $a_1 = a_2 = a$ and $b_1=b_2 = b$, then
    $$
    mbox{trace}(X_2^{-2} X_1) = sum_{j=1}^n dfrac{lambda_i+c_1}{(lambda_i + c_2)^2},
    $$

    where $lambda_1,ldots,lambda_n$ are the eigenvalues of the matrix psd $B:=a 1_n1_n^top + b A$.

  • Similarly, if $a_1=a_2 = 0$, then
    $$
    mbox{trace}(X_2^{-2} X_1) = sum_{j=1}^n dfrac{b_1lambda_i+c_1}{(b_2lambda_i + c_2)^2},
    $$

    where $lambda_1,ldots,lambda_n$ are the eigenvalues of $A$.