Finding Matrix A from Dynamics x'=Ax

[shaner-baner]

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.)

 

[neurocomp2003]

all you have is the D(x)=sum(x_i)? are you given x'?

 

[tacman]

To find the matrix A from the dynamics x'=Ax, we can use the data points collected through the scalar function D(x)=(sum of the components of the vector x). By using these data points, we can construct a system of equations that relates the values of x and D(x) to the matrix A. This system of equations can then be solved to determine the entries of A.

However, the uniqueness of the solution depends on the number of data points collected and the structure of the matrix A. If we have a sufficient number of data points, we can uniquely determine the entries of A. However, if we have fewer data points, there may be infinitely many solutions for the matrix A. This is because the system of equations may be underdetermined, meaning there are not enough equations to uniquely determine the matrix A.

The structure of the matrix A also plays a role in the uniqueness of the solution. If A has distinct eigenvalues, then we can determine the entries of A uniquely. However, if A has repeated eigenvalues, there may be infinitely many solutions for A. This is because the eigenvectors corresponding to repeated eigenvalues may be linearly dependent, making it impossible to determine the entries of A uniquely.

In summary, the ability to find the matrix A from the dynamics x'=Ax depends on the number of data points collected and the structure of the matrix A. With enough data points and a matrix with distinct eigenvalues, we can uniquely determine the entries of A. Otherwise, there may be infinitely many solutions for A.