Data
I have three $N \times N$ complex hermitian matrices $A=xx^{H}$,$R=rr^{H}$ and a positive-definite matrix $B$. Here $x$ and $r$ are two $N \times 1$ complex vectors. Let $\lambda_{i}, 1\leq i\leq N$ denotes the N eigenvalues of B which are also positive. Clearly $A$ and $R$ are two rank one positive semi-definite matrices. $B$ is invertible.
What I need to find
- What is the largest eigenvalue of the GEVP?
\begin{align} (A\otimes R)v=\gamma (B\otimes R)v \end{align}
- Will the maximum eigenvalue be (seemingly nice) $||r||^{2}x^{H}B^{-1}x$?
What I know
- Consider the generalized Eigenvalue problem (GEVP) \begin{align} Av=\gamma Bv \end{align} Since $B$ is invertible, this is equivalent to find the eigenvalues of $B^{-1}A$, in fact , since $A$ is rank one matrix, there is only one eigenvalue which will be positive, and it will be given by $x^{H}B^{-1}x$ ($A=xx^{H}$).
- Now I am interested in the matrices, $A \otimes R$ and $B \otimes R$ which are $N^{2} \times N^{2}$ in dimension. Now $A \otimes R$ is a rank one matrix, and its only non-zero eigenvalue is $||x||^{2}||r||^{2}$. $B \otimes R$ is a positive semi-definite matrix with $N$ of its eigenvalues being $\lambda_{i}||r||^{2}, 1\leq i \leq N$ and rest of the $N^{2}-N$ eigenvalues being zero.