Isospectral Flows for Second Order Systems

The lambda-matrix with complex matrix coefficients A_0, A_1, A_2,..., A_p defines a linear dynamic system of dimension mxn. When m=n, and when det(A(\lambda)) is non-zero for some values of \lambda, the eigenvalues of this system are well-defined. A one-parameter trajectory of such a system A_0(\sigma), A_1(\sigma),... is an isospectral flow if the eigenvalues and the dimensions of the associated eigenspaces are the same for all parameter values \sigma. This paper presents the most general form for isospectral flows of linear dynamic systems of orders p=2,3,4, and the forms for isospectral flows for even higher order systems are evident from the patterns emerging. Based on the definition of a class of coordinate transformations called structure-preserving transformations, the concept of isospectrality and the associated flows is seen to extend to cases where m does not equal n.