# linear algebra – How to find upper and lower bound

Let $$Sigma in S_{++}^n$$ be a symmteric positive definte matrix with all diagonal entries one. Let $$U in R^{n times k_1}$$, $$W in R^{n times k_2}$$, $$Lambda in R^{k_1 times k_1}$$ and $$T in R^{k_2 times k_2}$$, where $$Lambda$$ and $$T$$ are both diagonal matrix with positive elements, and $$n > k_2 > k_1$$. We also know $$text{trace}(mathbf{Lambda}) = mu times text{trace}(mathbf{T})$$ and sum of absolute values of all the elements of $$U$$ is less than $$W$$. Then how can I find upper and lower bound on-

begin{align*} frac{|Sigma – UTU^top|_F^2}{|Sigma – WLambda W^top|_F^2} end{align*}

in terms of $$mu$$, $$W$$, $$Lambda$$ and $$Sigma$$.