# $M \cdot [x,y,z]^t=[x^2,y^2,z^2]^t$ - can this be solved similarly like the eigenvector-problem?

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?

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