Dear mathematics community, I got a question like this:
$A,B$ are two square matrixes of order $n$, and $A=(B-\frac1 {110}E)^T(B+\frac1{110}E)$, prove that for any $n-$dimensional vector $\xi$, there exists at least one non-zero solution for the system of equations $A^T(A^2+A)X=A^T\xi$.
I don't see any hope in expending $A$, then the $A$ and $A^T$ reminded me of the least square solution , i.e., there exists at least one non-zero solution for the system of equations $A^T(A^2+A)X=A^T\xi $ if and only if there exists at least one non-zero least square solution for $(A+I)X=\xi$. So I need to prove that $\xi=0$ is not the least square solution for $(A+I)X=\xi$, then I was stuck.
Could anyone help me out here?