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?

  • $\begingroup$ Is your equation written with implicit sums? $\endgroup$ Commented Feb 18, 2023 at 6:57
  • $\begingroup$ 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. $\endgroup$
    – B K
    Commented Feb 18, 2023 at 7:37
  • $\begingroup$ And yes, a and b are summed over. $\endgroup$
    – B K
    Commented Feb 18, 2023 at 10:27
  • $\begingroup$ So your equation is one scalar linear equation in n^2 unknowns? $\endgroup$ Commented Feb 18, 2023 at 23:16
  • $\begingroup$ If by one scalar linear equation you mean both sides of the equation end up being scalars, yes. $\endgroup$
    – B K
    Commented Feb 19, 2023 at 15:05


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