Bounding $\frac{\operatorname{Tr}WC}{\operatorname{Tr}WCW^T}$ Suppose $C$ is a positive definite real-valued matrix and $W$ positive stable real-valued (all eigenvalues have positive real part). The following seems true empirically, is it possible to show this rigorously?
$$\frac{2\operatorname{Tr}WC}{\operatorname{Tr}WCW^T}\ge\lambda_{\text{min}}((W+W^T)(WW^T)^{-1})$$
Here $\lambda_{\text{min}}(A)$ refers to the smallest eigenvalue
When $W$ is positive definite, the equation above simplifies to
$$\frac{\operatorname{Tr}WC}{\operatorname{Tr}WCW^T}\ge \lambda_{\text{min}}W^{-1}$$
 A: Here's a proof for the statement where $W$ is positive definite. Hopefully this can help prove the more general statement although I don't see how to at the moment.
Consider the optimization
$$
\begin{aligned}
\min_C& \quad \frac{\mathrm{Tr}[W C]}{\mathrm{Tr}[W^2 C]} \\
\mathrm{s.t.}& \quad C \geq 0
\end{aligned}
$$
Notice that the objective function is invariant under rescaling of $C$. Therefore, for any feasible point we could always rescale $C$ so that $\mathrm{Tr}[W^2 C] = 1$ without changing the objective value. This implies that the optimization is equivalent to the following optimization
$$
\begin{aligned}
\min_C& \quad \mathrm{Tr}[W C] \\
\mathrm{s.t.}& \quad \mathrm{Tr}[W^2 C] = 1 \\
& \quad C \geq 0
\end{aligned}
$$
This is an semidefinite program it has a strictly feasible point (namely $C = W^{-2}/d$ where $d$ is the dimension) and hence its optimal value is equal to the optimal value of the dual problem
$$
\begin{aligned}
\max & \quad \mu \\
\mathrm{s.t.}& \quad W - \mu W^2  \geq 0
\end{aligned}
$$
But thinking about the spectral decomposition of $W$ this positive semidefinite constraint is just equivalent to $\lambda_i - \mu \lambda_i^2 \geq 0$ for all $i$ where $\lambda_i$ are the eigenvalues of $W$. Rearranging we find
$$
\mu \geq \frac{1}{\lambda_i} \qquad \text{ for all $i$}
$$
and hence $\mu \geq \lambda_{\min}(W^{-1})$ from which we have the result
$$
\frac{\mathrm{Tr}[W C]}{\mathrm{Tr}[W^2 C]} \geq \lambda_{\min}(W^{-1})\,.
$$
