g S to too and back again along the curves {otion/20<0<00 There is no error in doing so, but, due to periodicity of the eigenvalues, the net number of encirclements induced by tracing these trajectories will always be zero. 3.5 LPTV SYSTEM ANALYSIS USING HTMS 3.5.2.4 Computing the dominant HTM eigenvalues As was the case for HTM inversions, computing the eigenvalues of an co-dimensional HTM is a hard problem to solve. However, keeping in mind that we are only interested in eigenvalues that are sufficiently large, the problem can be simplified by exploiting the characteristics of open-loop gain HTMs as they occur in practice. In what follows, we discuss three methods for HTM eigenvalue computations. The methods parallel the ones introduced in sections 3.4.3.1-3.4.3.3 to compute the inverse of a HTM. Truncation-based eigenvalue computations Again, the most straightforward strategy is to truncate the open-loop gain HTM G(s), i.e. reduce it to a finite-dimensional matrix. Eigenvalues can then be computed using standard textbook algorithms [Gol96] . If the truncated matrix is very large, it is better to use subspace methods [Saad92, Sore00] to find the largest eigenvalues. Loops with a dominant LTI component A second approach towards estimating eigenvalue locations makes use of Gershgorin's theorem [Go196, Mac89]: Theorem (Gershgorin's theorem): Let Z = [Zk,1] be a complex matrix of dimensions M X M. Then the eigenvalues of Z lie in the union of the M (Gershgorin) circles, each with center Zk,k and radius (3.113) 1 # k As G(s) can be considered as the limit of a sequence of finite-dimensional matrices, we obtain that for each S EC and for each eigenvalue X(s) of G(s): (3.114) All eigenvalues X(s) of G(s) are hence contained within the Gershgorin circles with center points Gk,k(s) and radii Rk(s) = Gl,k(s) This theorem is especially useful when G(s) is a near-diagonal matrix, i.e. if the time-varying feedback loop contains a dominant LTI component. In that case, it roughly holds that (3.115) whereby the radius Rk(s) of