# Expression involving inverse of block matrices and matrix exponentials

I'm struggling to simplify $$B$$ which is given by $$B=\left(A^{-1}\right)^TS\left(A^{-1}\right)$$ with S a symmetric matrix of size $$2m \times 2m$$ and A a matrix given by $$A=\left[\begin{matrix} Ve^{\Lambda t_i} \\ Ve^{\Lambda t_f} \end{matrix}\right]$$ where $$V$$ is an $$m \times 2m$$ matrix and $$\Lambda$$ is a diagonal matrix of size $$2m \times 2m$$.

Any ideas? For example, can we write B by using 4 blocks of size $$m \times m$$?

In case it helps, $$A$$ can also be written as $$A=\left[\begin{matrix} V_1e^{\Lambda_1 t_i} & V_2e^{\Lambda_2 t_i}\\ V_1e^{\Lambda_1 t_f} & V_2e^{\Lambda_2 t_f} \end{matrix}\right]$$ where $$V_1 = V(:,1:m)$$ $$V_2 = V(:,m+1:2m)$$ $$\Lambda_1 = \Lambda(1:m,1:m)$$ $$\Lambda_2 = \Lambda(m+1:2m,m+1:2m)$$ and of course $$\Lambda_1$$ and $$\Lambda_2$$ are diagonal matrices.