Suppose we have a linear matrix inequality (aka LMI aka spectahedron aka linear matrix pencil):

$$A_{0}+x_{1}A_{1}+x_{2}A_{2}+\ldots+x_{m}A_{m} \succeq 0.$$

(The notation $X \succeq Y$ means that $X-Y$ is positive semidefinite).

If the LMI is feasible, then there is some $P \succeq 0$ so that $A_{0}+x_{1}A_{1}+x_{2}A_{2}+\ldots+x_{m}A_{m}=P$.

I would like to know if there is a way to express the $x_{i}$s as functions of $P$ and the $A_{i}$s.

A similar situation in which such a thing is possible is the Lyapunov equation $A^{T}X+XA=H$ for a diagonalizable $A=SDS^{-1}$ where one has: $$ X=(S^{-1})^{T}[L(A)] \circ (S^{T}HS)]S^{-1}, $$

with $L(A)=[(d_{i}+d_{j})^{-1}]$ and $\circ$ denoting the entrywise Schur matrix product. (Cf. pp. 300-301 in the book *Topics in Matrix Analysis* by Horn and Johnson for the derivation).