Given I have a matrix M, say 3x3, and invertible, and we want to determine the coefficients $[x,y,z]$ by the formula $$ M \cdot \begin{bmatrix} x\\y\\z \end{bmatrix} = \begin{bmatrix} x^2\\y^2\\z^2 \end{bmatrix} $$ If we write $X=diag(x,y,z) \qquad $ and $ U=[1,1,1]^t \qquad $ we have $$ M \cdot X \cdot U = X^2 \cdot U $$ and this reminded me remotely of the eigenvalue/eigenvector problem.
Q: Is there some nice procedure how to approach/solve this in general?
