# 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 $$A$$.

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