Isospectral flows that preserve matrix structure Original Research Article

The matrix A = (aij) set membership, variant Sn is said to lie on a strict undirected graph image if aij = 0 (i ≠ j) whenever (i, j) is not in image. If S is skew-symmetric, the isospectral flow image maintains the spectrum of A. We consider isospectral flows that maintain a matrix A(t) on a given graph image. We review known results for a graph image that is a (generalised) path, and construct isospectral flows for a (generalised) ring, and a star, and show how a flow may be constructed for a general graph. The analysis may be applied to the isospectral problem for a lumped-mass finite element model of an undamped vibrating system. In that context, it is important that the flow maintain other properties such as irreducibility or positivity, and we discuss whether they are maintained.