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Toeplitz and Circulant Matrices: A Review
Foundations and Trends® in
Communications and Information Theory
Volume 2 Issue 3
DOI: 10.1561/0100000006
Toeplitz and Circulant Matrices: A Review
Robert M. Gray
Stanford University, Stanford CA 94305, USA, rmgray@stanford.edu
Abstract
The fundamental theorems on the asymptotic behavior of eigenvalues, inverses, and products of banded Toeplitz matrices and
Toeplitz matrices with absolutely summable elements are derived in a tutorial manner. Mathematical elegance and generality
are sacrificed for conceptual simplicity and insight in the hope of making these results available to engineers lacking either
the background or endurance to attack the mathematical literature on the subject. By limiting the generality of the matrices
considered, the essential ideas and results can be conveyed in a more intuitive manner without the mathematical machinery
required for the most general cases. As an application the results are applied to the study of the covariance matrices and
their factors of linear models of discrete time random processes.
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