Suppose I have a symmetric matrix $A_{1000\times 1000}$, which can be represented by:
$A = J G J^T$
where $J$ in 1000x3 is full column rank dense matrix; $G$ in 3x3 is a nonsingular dense matrix.
What is the fastest way to obtain ONLY the maximum eigenvalue of $A$?
I know that the eigenvalue problem of symmetric matrix can be faster than that of general dense matrix, but can the following features of the problem make it even faster ?
only the maximum eigenvalue of $A$ is needed;
$A = J G J^T$, rank($A$) = 3, and $A$ has only 3 nonzero eigenvalues
Can $LDL^T$ decomposition work any good? I would prefer to implement it via Eigen C++.
Does $B=J^TJG$ has the same nonzero eigenvalues as $A$?