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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).

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The system is $A_0 + x_1A_1 + \ldots + x_mA_m = P$ is just a linear system in the $x_i$. So you could reformat the data to look like $Ax=b$ and then get $x = A^\dagger b$. Is there any reason to expect you could do any better than this? Without any special information about the $A_i$ this looks like a pretty generic linear system. – Noah Stein Jul 30 '13 at 12:44
@NoahStein The reason is that the Lyapunov equation does have a matricial formula which is not at all evident from the scalar formulation. But that's all the evidence I have... (btw, $x=A^{\dagger}b$ does not always hold, you also need to have $x \perp null(A)$, ifirc). – Felix Goldberg Jul 30 '13 at 12:47
But in the Lyapunov case the difficulty is that you're looking for a positive semidefinite solution whereas here the $x_i$ are otherwise unconstrained, right? (And yes, with the pseudoinverse I was assuming that $Ax=b$ has a solution.) – Noah Stein Jul 30 '13 at 13:33
@NoahStein Actually, I do need the $x_{i}$s to be positive, but I was planning to deal with that later... :) – Felix Goldberg Jul 30 '13 at 15:19

I would say that the answer is NO, since your problem is basically the general semidefinite program. There are obviously tractable special cases (like, if $A_1$ is psd, then you can set $x_i = \delta_{1i}, P=A_1.$)

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