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

$Avv^TAw=\lambda_1v+\lambda_2w$

$Aww^TAv=\lambda_1w+\lambda_2v$

$v^Tw=w^Tv=0$

$v^Tv=w^Tw=1$

where $\lambda_1, \lambda_2, \lambda_3$ are real. $A$ is a symmetric matrix.

How would you generalize to the case

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

$Aww^TAv+Bvv^TBw=\lambda_1w+\lambda_2v$Aww^TAv+Bww^TBv=\lambda_1w+\lambda_2v$Where both A and B are symmetric? Would it help if they are also similar and each of them has exactly$n/2$eigenvalues equal to$+1$and$n/2$eigenvalues equal to$-1$? 4 edited body Solve the following nonlinear equations for$v$and$wAvv^TAw=\lambda_1v+\lambda_2wAww^TAv=\lambda_1w+\lambda_3v$Aww^TAv=\lambda_1w+\lambda_2v$

$v^Tw=w^Tv=0$

$v^Tv=w^Tw=1$

where $\lambda_1, \lambda_2, \lambda_3$ are real. $A$ is a symmetric matrix.

How would you generalize to the case

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

$Aww^TAv+Bvv^TBw=\lambda_1w+\lambda_3v$Aww^TAv+Bvv^TBw=\lambda_1w+\lambda_2v$Where both A and B are symmetric? Would it help if they are also similar and each of them has exactly$n/2$eigenvalues equal to$+1$and$n/2$eigenvalues equal to$-1$? 3 added 289 characters in body; added 12 characters in body Solve the following nonlinear equations for$v$and$wAvv^TAw=\lambda_1v+\lambda_2wAww^TAv=\lambda_1w+\lambda_3vv^Tw=w^Tv=0v^Tv=w^Tw=1$where$\lambda_1, \lambda_2, \lambda_3$are real.$A$is a symmetric matrix. How would you generalize to the case$Avv^TAw+Bvv^TBw=\lambda_1v+\lambda_2wAww^TAv+Bvv^TBw=\lambda_1w+\lambda_3v$Where both A and B are symmetric? Would it help if they are also similar and each of them has exactly$n/2$eigenvalues equal to$+1$and$n/2$eigenvalues equal to$-1\$?

2 edited body
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