Toeplitz and circulant matrices: A review

Foundations and Trends in Communications and Information Theory

Volumn 2, Issue 3, 2006, Pages 155-239

  Gray, Robert M ^a  

^a STANFORD UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EIGENVALUES AND EIGENFUNCTIONS; ENGINEERS; LINEAR EQUATIONS; MATHEMATICAL MODELS; MATRIX ALGEBRA; RANDOM PROCESSES;

COVARIANCE MATRICES; MATHEMATICAL LITERATURES; TOEPLITZ;

INFORMATION THEORY;

EID: 33645776880     PISSN: 15672190     EISSN: 15672328     Source Type: Journal    
DOI: 10.1561/0100000006     Document Type: Review

References (34)

1. G. Baxter, "An Asymptotic Result for the Finite Predictor," Math. Scand., vol. 10, pp. 137-144, 1962.
2. G. Baxter, "A Norm Inequality for a 'Finite-Section' Wiener-Hopf Equation," Illinois J. Math., pp. 97-103, 1962.
3. T. Berger, Rate Distortion Theory: A Mathematical Basis for Data Compression. Prentice Hall, Englewood Cliffs, New Jersey, 1971.
4. A. Böttcher and S. Grudsky, Toeplitz Matrices, Asymptotic Linear Algebra, and Functional Analysis. Birkhäuser, 2000.
5. A. Böttcher and B. Silbermann, Introduction to Large Truncated Toeplitz Matrices. Springer, New York, 1999.
6. W. Cheney, Introduction to Approximation theory. McGraw-Hill, 1966.
7. T. A. Cover and J. A. Thomas, Elements of Information Theory. Wiley, New York, 1991.
8. P. J. Davis, Circulant Matrices. Wiley-lnterscience, NY, 1979.
9. D. Fasino and P. Tilli, "Spectral clustering properties of block multilevel Hankel matrices," Linear Algebra and its Applications, vol. 306, pp. 155-163, 2000.
10. R. M. Gray and L. D. Davisson, An Introduction to Statistical Signal Processing. Cambridge University Press, London, 2005.
11. R. M. Gray, "On Unbounded Toeplitz Matrices and Nonstationary Time Series with an Application to Information Theory," Information and Control, vol. 24, pp. 181-196, 1974.
12. R. M. Gray, "Information Rates of Autoregressive Processes," IEEE Trans. on Info. Theory, vol. IT-16, no. 4, pp. 412-421, July 1970.
13. R. M. Gray, "On the asymptotic eigenvalue distribution of Toeplitz matrices," IEEE Transactions on Information Theory, vol. 18, pp. 725-730, November 1972.
14. U. Grenander and M. Rosenblatt, Statistical Analysis of Stationary Time Series. Wiley and Sons, NY, 1966. Chapter 1.
15. U. Grenander and G. Szegö, Toeplitz Forms and Their Applications. University of Calif. Press, Berkeley and Los Angeles, 1958.
16. A. J. Huffman and H. W. Wielandt, "The variation of the spectrum of a normal matrix," Duke Math. J., vol. 20, pp. 37-39, 1953.
17. T. Kailath, A. Sayed, and B. Hassibi, Linear Estimation. Prentice Hall, New Jersey, 2000.
18. P. Lancaster, Theory of Matrices. Academic Press, NY, 1969.
19. J. Pearl, "On Coding and Filtering Stationary Signals by Discrete Fourier Transform," IEEE Trans. on Info. Theory, vol. IT-19, pp. 229-232, 1973.
20. C. L. Rino, "The Inversion of Covariance Matrices by Finite Fourier Transforms," IEEE Trans. on Info. Theory, vol. IT-16, no. 2, pp. 230-232, March 1970.
21. J. Rissanen and L. Barbosa, "Properties of Infinite Covariance Matrices and Stability of Optimum Predictors," Information Sciences, vol. 1, pp. 221-236, 1969.
22. W. Rudin, Principles of Mathematical Analysis. McGraw-Hill, NY, 1964.
23. W. F. Trench, "Asymptotic distribution of the even and odd spectra of real symmetric Toeplitz matrices," Linear Algebra Appl., vol. 302-303, pp. 155-162, 1999.
24. W. F. Trench, "Absolute equal distribution of families of finite sets," Lin. Alg. Appl., vol. 367, pp. 131-146, 2003.
25. W. F. Trench, "Absolute equal distribution of the spectra of Hermitian matrices," Lin. Alg. Appl., vol. 366, pp. 417-431, 2003.
26. W. F. Trench, "A note on asymptotic zero distribution of orthogonal polynomials," Lin. Alg. Appl., vol. 375, pp. 275-281, 2003.
27. W. F. Trench, "Simplification and strengthening of Weyl's definition of asymptotic equal distribution of two families of finite sets," Cubo A Mathematical Journal, vol. 06, no. 3, pp. 47-54, 2004.
28. W. F. Trench, "Absolute equal distribution of the eigenvalues of discrete Sturm-Liouville problems," J. Math. Anal. Appl., in press.
29. B. S. Tsybakov, "Transmission capacity of memoryless Gaussian vector channels (in Russian)," Probl. Peredach. Inform., vol. 1, pp. 26-40, 1965.
30. B. S. Tsybakov, "On the transmission capacity of a discrete-time Gaussian channel with filter (in Russian)," Probl. Peredach. Inform., vol. 6, pp. 78-82, 1970.
31. E. E. Tyrtyshnikov, "Influence of matrix operations on the distribution of eigenvalues and singular values of Toeplitz matrices," Linear Algebra and its Applications, vol. 207, pp. 225-249, 1994.
32. E. E. Tyrtyshnikov, "A unifying approach to some old and new theorems on distribution and clustering," Linear Algebra and its Applications, vol. 232, pp. 1-43, 1996.
33. H. Widom, "Toeplitz Matrices," in Studies in Real and Complex Analysis, (J. I.I. Hirschmann, ed.), MAA Studies in Mathematics, Prentice-Hall, Englewood Cliffs, NJ, 1965.
34. J. H. Wilkinson, "Elementary proof of the Wielandt-Hoffman theorem and of its generalization," Tech. Rep., Stanford University, Department of Computer Science, January 1970. Report Number CS-TR-70-150.