Is it possible to find the closed form expression for the largest eigenvalue of the following matrix?

As we know, a matrix multiplied by a diagonal matrix can be viewed as a scaling of itself. And multiplied by a unitary matrix can be viewed as a rotation from the original matrix. So now I'm wondering how to analytically find the largest eigenvalue of the following mentioned matrix.

Could anyone please give any inspirations? Many thanks!

$\mathbf{M} = \mathbf{\Lambda_G V \Lambda_p V}^H\mathbf{\Lambda_G^H}$, where $\mathbf{\Lambda_G}$ is a complex diagonal matrix, and $\mathbf{\Lambda_p}$ is real and its diagonal elements are all non-negative. $\mathbf{V}$ is a unitary matrix such that $\mathbf{VV}^H=\mathbf{I}$.

The problem discription

• So $\Lambda_P$ is real? "Non-negative" does not make any sense for complex matrices. – NickD Jul 6 '18 at 13:27
• Sorry, my mistake... already fixed it. Thanks for pointing out! – R.F. Jul 6 '18 at 13:31