Least squares realization of LTI models is an eigenvalue problem

We show how least squares optimal realization of autonomous linear time-invariant dynamical systems from given data, reduces to the solution of an eigenvalue problem. In this short paper, we can only schematically sketch the different steps: The first order optimality conditions result in a multi-parameter eigenvalue problem. The eigenvalue $n$-tuples are calculated from the null space of a quasi-Toeplitz block Macaulay matrix, which is shown to be multishift-invariant. This last property is then exploited via nD ‘exact’ realization theory, leading through several eigenvalue problems to the optimal model parameters.