## Enforcing solvability of a nonlinear matrix equation and estimation of multivariate ARMA time series

Please always quote using this URN: urn:nbn:de:0296-matheon-12409

- The matrix equation $X+AX^{-1}A^T=B$, arising in parameter estimation of certain time series models, is solvable only for certain values of the matrices $A,B$. We present a numerical method to modify $A,B$ in order to make the matrix equation solvable. Since solvability depends on the location of the eigenvalues of the palindromic matrix polynomial $\lambda^2 A+\lambda B+A^T$, our method works by moving those eigenvalues to specified locations using first order spectral perturbation theory. The method is heuristic but works in practice, as is supported by several compelling numerical examples. These examples arise from parameter estimation of a common time series model, the multivariate ARMA(1,1).