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 $Sigma$.