I have a system of equations that can be written as
${(\bf{A}} + \lambda{\bf{B}}){\bf{x}} = 0$
Where ${\bf{A}}$ and ${\bf{B}}$ are $n \times m$, integer matrices. I know that there are several values of $\lambda$ that correspond with a nontrivial $x$ and that solve the system. What can I do to find $\lambda$ in general?
My first idea was to just do Gaussian Elimination on the matrix ${\bf{C}} ={(\bf{A}} + \lambda{\bf{B}})$, essentially making it square. However, it seems like I am throwing away information if I do that.
Claim: For a fixed $A,B, \lambda$, $$ (A + \lambda B)x = 0 \iff (A + \lambda B)^T(A + \lambda B)x = 0 $$ Proof: $\implies$ holds trivially. To prove the other implication, note that $$ (A + \lambda B)^T(A + \lambda B)x = 0 \implies\\ x^T(A + \lambda B)^T(A + \lambda B)x = 0 \implies\\ [(A + \lambda B)x]^T[(A + \lambda B)x] = 0 \implies\\ \|(A + \lambda B)x\|^2 = 0 \implies\\ (A + \lambda B)x = 0 $$ Thus, as you indicate, it is sufficient to find the $\lambda$ such that $$ \det[(A + \lambda B)^T(A + \lambda B)] = 0 $$