General isospectral flows for linear dynamic systems Original Research Article

The λ-matrix A(λ)=(A0+λA1+λ2A2+cdots, three dots, centeredλkAkcdots, three dots, centered+λlAl) with matrix coefficients image defines a linear dynamic system of dimension (m×n). When m=n, and when det(A(λ))≠0 for some values of λ, the eigenvalues of this system are well-defined. A one-parameter trajectory of such a system {A0(σ),A1(σ),A2(σ)…Aℓ(σ)} is an isospectral flow if the eigenvalues and the dimensions of the associated eigenspaces are the same for all parameter values σset membership, variantIR. This paper presents the most general form for isospectral flows of linear dynamic systems of orders (ℓ=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≠n).