4,508 research outputs found
Companion Matrices and Their Relations to Toeplitz and Hankel Matrices
In this paper we describe some properties of companion matrices and
demonstrate some special patterns that arise when a Toeplitz or a Hankel matrix
is multiplied by a related companion matrix. We present a new condition,
generalizing known results, for a Toeplitz or a Hankel matrix to be the
transforming matrix for a similarity between a pair of companion matrices. A
special case of our main result shows that a Toeplitz or a Hankel matrix can be
extended using associated companion matrices, preserving the Toeplitz or Hankel
structure respectively
The use of the QR factorization in the partial realization problem
The use of the QR factorization of the Hankel matrix in solving the partial realization problem is analyzed. Straightforward use of the QR factorization results in a realization scheme that possesses all of the computational advantages of Rissanen's realization scheme. These latter properties are computational efficiency, recursiveness, use of limited computer memory, and the realization of a system triplet having a condensed structure. Moreover, this scheme is robust when the order of the system corresponds to the rank of the Hankel matrix. When this latter condition is violated, an approximate realization could be determined via the QR factorization. In this second scheme, the given Hankel matrix is approximated by a low-rank non-Hankel matrix. Furthermore, it is demonstrated that column pivoting might be incorporated in this second scheme. The results presented are derived for a single input/single output system, but this does not seem to be a restriction
Detecting optimality and extracting solutions in polynomial optimization with the truncated GNS construction
A basic closed semialgebraic subset of is defined by
simultaneous polynomial inequalities . We
consider Lasserre's relaxation hierarchy to solve the problem of minimizing a
polynomial over such a set. These relaxations give an increasing sequence of
lower bounds of the infimum. In this paper we provide a new certificate for the
optimal value of a Lasserre relaxation be the optimal value of the polynomial
optimization problem. This certificate is that a modified version of an optimal
solution of the Lasserre relaxation is a generalized Hankel matrix. This
certificate is more general than the already known certificate of an optimal
solution being flat. In case we have optimality we will extract the potencial
minimizers with a truncated version of the Gelfand-Naimark-Segal construction
on the optimal solution of the Lasserre relaxation. We prove also that the
operators of this truncated construction commute if and only if the matrix of
this modified optimal solution is a generalized Hankel matrix. This
generalization of flatness will bring us to reprove a result of Curto and
Fialkow on the existence of quadrature rule if the optimal solution is flat and
a result of Xu and Mysovskikh on the existance of a Gaussian quadrature rule if
the modified optimal solution is generalized Hankel matrix. At the end, we
provide a numerical linear algebraic algorithm for dectecting optimality and
extracting solutions of a polynomial optimization problem
Fast and Provable Algorithms for Spectrally Sparse Signal Reconstruction via Low-Rank Hankel Matrix Completion
A spectrally sparse signal of order is a mixture of damped or
undamped complex sinusoids. This paper investigates the problem of
reconstructing spectrally sparse signals from a random subset of regular
time domain samples, which can be reformulated as a low rank Hankel matrix
completion problem. We introduce an iterative hard thresholding (IHT) algorithm
and a fast iterative hard thresholding (FIHT) algorithm for efficient
reconstruction of spectrally sparse signals via low rank Hankel matrix
completion. Theoretical recovery guarantees have been established for FIHT,
showing that number of samples are sufficient for exact
recovery with high probability. Empirical performance comparisons establish
significant computational advantages for IHT and FIHT. In particular, numerical
simulations on D arrays demonstrate the capability of FIHT on handling large
and high-dimensional real data
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