Finding the closest Toeplitz matrix

Abstract

Abstract. The constrained least-squares n × n-matrix problem where the feasibility set is the subspace of the Toeplitz matrices is analyzed. The general, the upper and lower triangular cases are solved by making use of the singular value decomposition. For the symmetric case, an algorithm based on the alternate projection method is proposed. The implementation does not require the calculation of the eigenvalue of a matrix and still guarantees convergence. Encouraging preliminary results are discussed.Mathematical subject classification: 65F20, 65F30, 65H10.

Robust Stopping Criteria for Dykstra's Algorithm

“…The solution x * is called the projection of x 0 onto Ω and is denoted by P Ω (x 0 ). Dykstra's algorithm for solving (1) has been extensively studied since it fits in many different applications (see [1,2,4,8,9,11,12,13,18,21,23,24,26,28,29]). Here, we consider the case…”

The inverse eigenvalue problem for a Hermitian reflexive matrix and the optimization problem

“…The interest of inverse eigenvalue problems is remarkable for their applications in engineering [1,2,6,7,12,15]. Recently, some specific problems have been tackled [9,10,11,14].…”

“…By a similar reasoning as before, the form of the matrix A is given by (4). Substituting partition (6) in AX = XD, it is obtained that…”

“…The inverse eigenvalue problem for Hermitian reflexive (antireflexive) matrices with respect to a Hermitian tripotent matrix was analyzed in [9]. Also, for structured matrices such as Toeplitz and generalized K-centrohermitian, problems like these ones have been studied in [6,7]. For a left and right inverse eigenvalue problem with reflections we can refer to [11].…”

Optimización del problema de valor propio inverso para matrices estructuradas

“…Con técnicas similares, G. Eberle y M. C. Maciel consideraron en [12] el problema de optimización de mínimos cuadrados restringido a matrices Toeplitz, matrices Toeplitz triangulares y matrices Toeplitz simétricas. Este tema fue abordado nuevamente en [47] por J. Yang y Y. Deng transformando el problema original en una forma cuadrática, y obteniendo la solución mediante la resolución de un sistema lineal de ecuaciones.…”