Schur parametrization of symmetric matrices with any rank profile

The conceptual solution to the parametrization problem for symmetric indefinite matrices P is addressed. Beyond the fact that the symmetric matrix to be parametrized may have positive, negative and vanishing eigenvalues, it may as well comprise singular leading submatrices. For the parametrization, the lossless inverse scattering (LIS) framework is employed, which amounts to the mapping of a given symmetric matrix P onto a lossless and cascaded model structure. This leads to a recursive algorithm for the identification of the model parameters, the so-called Schur parameters, which turn out to form a set of vector-valued quantities to determine the individual lossless layers in the LIS model