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I crossposted this question on math.stackexchange.

Background: Suppose that an investor's utility is both determined by the state and her action taken. A fact of life is that she can't observe the realized state. However, she can deduce the probability of each state conditional on different signals. One natural question is whether the investor will be better off if signals are more informative. Contrasted with game theoretical setting, Blackwell's theorem answers this question affirmatively.

An information system relates signals to states by assigning probabilities. Let's say states be indexed by a finite set $I$, signals be indexed by another finite set $J$. $Q$ is a $|I| \times |J|$ matrix. $Q_{ij}$ represents the probability that at the $i$th state, the $j$th signal displays. Thus $Q$ is a Markovian matrix.

For two information systems $M$, $N$, we say $M$ is not less informative than $N$ if there exists a matrix $U$ with appropriate dimension such that $MU = N$.

Question: Is there a way to justify this characterization other than appealing to Blackwell's theorem which seems to be tautological?

I tried some simple square matrices. For example, it matches intuition that identity matrix is the maximal informative matrix. At least in some situations the inverse of a invertible Markovian matrix could be not Markovian. But I don't find them convincing enough.

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