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 $\mathbf{VV}^H=\mathbf{I}$.
It is possible when the matrices involved are 2 x 2. See here for closed form solutions to 2 x 2 SVDs, and for the problem posed here, the singular values and the eigenvalues are the same thing. Likely doable for 3 x 3, since that should reduce to solving a third order polynomial, and there are (very messy) closed form solutions for doing that. Beyond that, good luck - you have entered the mathematics wastelands...