Abstract
Assuming that a ij is distributed uniformly in [—1,1] and a ii = 1, we compute the probability that a symmetric matrix A = [a ij ] 171-1 j=1 is positive semidefinite. The probability is also computed if A is a Toeplitz matrix. Finally, some results for partial matrices are presented.
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.
References
Arsene, G., Ceauşescu, Z., Constantinescu, T.: Schur analysis of some completion problems; Linear Algebra Appl. 109 (1988), 1–35.
Bakonyi, M., Constantinescu, T.: Schur’s Algorithm and Several Applications; Pitman Res. Notes Math. Ser. 261, Longman Scientific & Technical, Essex 1992.
Constantinescu, T.: Schur analysis of positive block-matrices; in: I.C. Gohberg (ed.): I. Schur Methods in Operator Theory and Signal Processing, Operator Theory: Adv. Appl. 18, Birkhäuser Verlag, Basel 1986, 191–206.
Foias, C., Frazho, A.: The Commutant Lifting Approach to Interpolation Problems; Operator Theory: Adv. Appl. 44, Birkhäuser Verlag, Basel 1990.
Fritzsche, B., Kirstein, B.: A Schur parametrization of non-negative hermitian and contractive block matrices and the corresponding maximum entropy problems; Optimization 19:5 (1988), 737–755.
Grone, B., Johnson, C.R., Sá, E.M., Wolkowicz, H.: Positive definite completions of partial hermitian matrices; Linear Algebra Appl. 58 (1984), 109–124.
James, G.: Modern Engineering Mathematics; Second Edition; Addison-Wesley Longman, Harlow 1996.
Johnson, C.R.: Matrix completion problems: a survey; in: C.R. Johnson (ed.): Matrix Theory and Applications, Proc. Symp. Appl. Math. (1990), 171–198.
Johnson, C.R., Lundquist, M., Nævdal, G.: Positive definite Toeplitz completions; J. London Math. Soc. (to appear).
Laurent, M., Poljak, S.: On a positive semidefinite relaxation of the cut polytope; Linear Algebra Appl. 223/224 (1995), 439–461.
Næevdal, G.: On a generalization of the trigonometric moment problem; Linear Algebra Appl. 258 (1997), 1–18.
Schur, I.: Öber Potenzreihen, die im Innern des Einheitskreises beschränkt sind; J. Reine Angew. Math. 147 (1917), 205–232.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer Basel AG
About this paper
Cite this paper
Johnson, C.R., Nævdal, G. (1998). The Probability that a (partial) matrix is positive semidefinite. In: Gohberg, I., Mennicken, R., Tretter, C. (eds) Recent Progress in Operator Theory. Operator Theory Advances and Applications, vol 103. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8793-9_10
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8793-9_10
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9776-1
Online ISBN: 978-3-0348-8793-9
eBook Packages: Springer Book Archive