Finding Matrix A from Dynamics x'=Ax

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The discussion focuses on determining the matrix A from the dynamics represented by the equation x' = Ax, where x is a vector and A is a matrix. The challenge arises from the inability to measure x directly, relying instead on the scalar function D(x) = sum(x_i). The consensus is that while it is possible to infer properties of matrix A from sufficient data points, the uniqueness of its entries is not guaranteed, leading to potentially infinite solutions. The structure of matrix A, particularly the nature of its eigenvalues, significantly influences the conclusions that can be drawn.

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shaner-baner
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I am working with a system governed by the dynamics x'=Ax (prime denotes differentiation w/ respect to t) where x is a vector and A is a matrix. Given the way our data is collected we can't measure x directly but rather a scalar function D(x)=(sum of the components of the vector x). My question is: given sufficiently many data points what can we conclude about the matrix A? Can we find its entries uniquely or are we stuck with a set of infinitely many solutions? And finally how does the answer to this question depend on the structure of the matrix (distinct/repeated eigenvalues etc.)
 
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all you have is the D(x)=sum(x_i)? are you given x'?
 

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