How can I tell if a tensor equation has a non-trivial solution or no solution at all?

I have the equation $$C^{ab}A_{ab}=X^{ab}(A_{ab}+B_{ab})$$. Is there a non-trivial solution to solve for the tensor $$X^{ab}$$ where none of the tensors can be set to zero? How would I solve for it?

If I say $$D_{ab}:=(A_{ab}+B_{ab})$$ is invertible then I can solve for $$X^{ab}$$. How can I solve for $$X^{ab}$$ if $$D_{ab}$$ or its constituents are not invertible?

• Is your equation written with implicit sums? Commented Feb 18, 2023 at 6:57
• Let's say $A_{ab}(x)$ and $B_{ab}(x*y)$ and $C^{ab}(z,y)$. Where x, y, and z are independent and x*y is the product of x and y variables.
– B K
Commented Feb 18, 2023 at 7:37
• And yes, a and b are summed over.
– B K
Commented Feb 18, 2023 at 10:27
• So your equation is one scalar linear equation in n^2 unknowns? Commented Feb 18, 2023 at 23:16
• If by one scalar linear equation you mean both sides of the equation end up being scalars, yes.
– B K
Commented Feb 19, 2023 at 15:05