Question

Solve the following nonlinear equations for $v$ and $w$

$Avv^TAw+Bvv^TBw=\lambda_1v+\lambda_2w$

$Aww^TAv+Bww^TBv=\lambda_1w+\lambda_2v$

$v^Tw=w^Tv=0$

$v^Tv=w^Tw=1$

where $\lambda_1, \lambda_2, \lambda_3$ are real. $A$, $B$ are n-by-n symmetric matrices. Furthermore, they are similar and each of them has exactly $n/2$ eigenvalues equal to $+1$ and $n/2$ eigenvalues equal to $-1$.

It is part of my attempt to minimize

$\sum_A |w^TAv|^2$

by Lagrange multiplier. Here A are all tensor products of Pauli matrices

Answer 1

Only a partial answer:

Suppose $\alpha:=v^TAw\neq 0$, $\beta:=v^T B w \neq 0$. Then the equations are $$(\alpha A + \beta B)v=\lambda_1v+\lambda_2w, $$ $$ (\alpha A + \beta B)w=\lambda_2v+\lambda_1w. $$ The idea in the answer to your other question still applies, you can tell that $v$ and $w$ are linear combinations of two eigenvectors $x_1,x_2$ of $\alpha A +\beta B$. I am not sure that this is enough in this case though, since getting the eigenpairs of $\alpha A + \beta B$ for yet-unknown $\alpha,\beta$ is not as easy as for a known matrix.