Abstract
This chapter is a positive matrix version of Chapter IV. First we give a complete characterization of all 2 by 2 and 3 by 3 positive block matrices. Then this is used to obtain some standard results for positive Toeplitz matrices and the Levinson algorithm. We also show that there is a one to one correspondence between the set of all n by n positive block matrices, and a set of n positive operators combined with n(n−1)/2 contractions. As a special case of this result, we also establish the fact that there is a one to one correspondence between the set of all positive block Toeplitz matrices on l 2n (H), and the set of all choice sequences {Γi} n−11 initiated on H.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Notes And Comments
Sz.-Nagy, B., Un calcul fonctionnel pour les contractions.–Sur la structure des dilatations unitaires des operateurs de l’espace de Hilbert, Seminari dell’Istituto Nazionale de Alta Matematica 1962–63, pp. 525–534.
Gohberg, I., and S. Goldberg, Basic Operator Theory, Birkhauser, Boston, 1981.
Devinatz, A. and M. Shinbrot, General Wiener-Hopf operators, Trans. American Math. Soc.,145 (1969) pp. 467–494.
Durszt, E. and B. Sz.-Nagy, Remarks on a paper of A. E. Frazho, Models for noncommuting operators, J. Functional Analysis, 52 (1983) pp. 146–147.
Morf, M., Vieira, A. and T. Kailath, Covariance characterization by partial autocorrelation matrices, The Annals of Statistics, 6 (1978) pp. 643–648.
Wiggins, R.A. and E.A., Robinson, Recursive solution to the multichannel filtering problem, J. Geophys. Res., 70 (1965) pp. 1885–1891.
Frazho, A.E. and D.K. Lee A pole placement procedure via H°° optimization, The 26 Annual Allerton Conference, Sept. (1988) pp. 872–880.
Constantinescu, T., Schur analysis of positive block matrices, I. Schur Methods in Operator Theory and Signal Processing; Operator Theory: Advances and Applications, 18, Ed. I. Gohberg (1986) pp. 191–206.
Arsene, Gr. and Z. Ceausescu, On intertwining dilations IV, Tokoku Math. J., 30 (1978) pp. 423–438.
Masani, P., Shift invariant spaces and prediction theory, Acta Math., 107 (1962) pp. 275290.
Masani, P., On the representation theorem of scattering, Bull. Amer. Math., 74 (1968) pp. 618–624.
Lev-Ari, H. and T. Kailath, Lattice filter parameterization and modeling of nonstationary processes, IEEE Trans. on Information Theory, 30 (1984) pp. 2–16.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1990 Springer Basel AG
About this chapter
Cite this chapter
Foias, C., Frazho, A.E. (1990). Positive Definite Block Matrices. In: The Commutant Lifting Approach to Interpolation Problems. OT 44 Operator Theory: Advances and Applications, vol 44. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7712-1_16
Download citation
DOI: https://doi.org/10.1007/978-3-0348-7712-1_16
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-7714-5
Online ISBN: 978-3-0348-7712-1
eBook Packages: Springer Book Archive