51,843 research outputs found
J-spectral factorization and equalizing vectors
For the Wiener class of matrix-valued functions we provide necessary and sufficient conditions for the existence of a -spectral factorization. One of these conditions is in terms of equalizing vectors. A second one states that the existence of a -spectral factorization is equivalent to the invertibility of the Toeplitz operator associated to the matrix to be factorized. Our proofs are simple and only use standard results of general factorization theory. Note that we do not use a state space representation of the system. However, we make the connection with the known results for the Pritchard-Salamon class of systems where an equivalent condition with the solvability of an algebraic Riccati equation is given. For Riesz-spectral systems another necessary and sufficient conditions for the existence of a -spectral factorization in terms of the Hamiltonian is added
Polynomial solutions to Hâ problems
The paper presents a polynomial solution to the standard Hâ-optimal control problem. Based on two polynomial J-spectral factorization problems, a parameterization of all suboptimal compensators is obtained. A bound on the McMillan degree of suboptimal compensators is derived and an algorithm is formulated that may be used to solve polynomial J-spectral factorization problems
A Spectral Theory for Tensors
In this paper we propose a general spectral theory for tensors. Our proposed
factorization decomposes a tensor into a product of orthogonal and scaling
tensors. At the same time, our factorization yields an expansion of a tensor as
a summation of outer products of lower order tensors . Our proposed
factorization shows the relationship between the eigen-objects and the
generalised characteristic polynomials. Our framework is based on a consistent
multilinear algebra which explains how to generalise the notion of matrix
hermicity, matrix transpose, and most importantly the notion of orthogonality.
Our proposed factorization for a tensor in terms of lower order tensors can be
recursively applied so as to naturally induces a spectral hierarchy for
tensors.Comment: The paper is an updated version of an earlier versio
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