Bounds on the extreme eigenvalues of positive-definite Toeplitz matrices

1988 ◽  
Vol 34 (2) ◽  
pp. 352-355 ◽  
Author(s):  
A. Dembo
Keyword(s):  
Toeplitz Matrices ◽  
Positive Definite ◽  
Extreme Eigenvalues

On Projection of a Positive Definite Matrix on a Cone of Nonnegative Definite Toeplitz Matrices

2018 ◽  
Vol 33 ◽  
pp. 74-82 ◽  
Author(s):  
Katarzyna Filipiak ◽  
Augustyn Markiewicz ◽  
Adam Mieldzioc ◽  
Aneta Sawikowska
Keyword(s):  
Toeplitz Matrices ◽  
Positive Definite Matrix ◽  
Positive Definiteness ◽  
Positive Definite ◽  
Asymptotic Cone ◽  
The Matrix ◽  
Multi Level ◽  
Nonnegative Definite ◽  
Unknown Covariance ◽  
Numer Math

We consider approximation of a given positive definite matrix by nonnegative definite banded Toeplitz matrices. We show that the projection on linear space of Toeplitz matrices does not always preserve nonnegative definiteness. Therefore we characterize a convex cone of nonnegative definite banded Toeplitz matrices which depends on the matrix dimensions, and we show that the condition of positive definiteness given by Parter [{\em Numer. Math. 4}, 293--295, 1962] characterizes the asymptotic cone. In this paper we give methodology and numerical algorithm of the projection basing on the properties of a cone of nonnegative definite Toeplitz matrices. This problem can be applied in statistics, for example in the estimation of unknown covariance structures under the multi-level multivariate models, where positive definiteness is required. We conduct simulation studies to compare statistical properties of the estimators obtained by projection on the cone with a given matrix dimension and on the asymptotic cone.

ON THE POSITIVENESS OF A FUNCTIONAL SYMMETRIC MATRIX USED IN DIGITAL FILTER DESIGN

2004 ◽  
Vol 13 (05) ◽  
pp. 1105-1110 ◽  
Author(s):  
YAN WU
Keyword(s):  
Signal Processing ◽  
Generating Function ◽  
Digital Filter ◽  
Simple Proof ◽  
Filter Design ◽  
Symmetric Matrix ◽  
Toeplitz Matrices ◽  
Positive Definite ◽  
Real Numbers ◽  
Digital Filter Design

This paper gives a simple proof for the positiveness of two important symmetric Toeplitz matrices used in communication and signal processing. It utilizes the shifting property of a so-called Uniformly Band-Restricted (UBR) function, which is the generating function for a generic functional symmetric matrix. It is shown that the functional symmetric matrix is positive definite if the UBR function is evaluated at a sequence of distinct real numbers.

scholarly journals Norms of positive definite Toeplitz matrices and detection of almost periodic components in random signals

2014 ◽  
pp. 861-875 ◽  
Author(s):  
Vadym Adamyan ◽  
José Luis Iserte ◽  
Igor M. Tkachenko ◽  
Gumersindo Verdú
Keyword(s):  
Toeplitz Matrices ◽  
Positive Definite ◽  
Almost Periodic ◽  
Random Signals ◽  
Periodic Components
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