Abstract
In a recent article Gowda and Sznajder (Linear Algebra Appl 432:1553–1559, 2010) studied the concept of Schur complement in Euclidean Jordan algebras and described Schur determinantal and Haynsworth inertia formulas. In this article, we establish some more results on the Schur complement. Specifically, we prove, in the setting of Euclidean Jordan algebras, an analogue of the Crabtree-Haynsworth quotient formula and show that any Schur complement of a strictly diagonally dominant element is strictly diagonally dominant. We also introduce the concept of Schur product of a real symmetric matrix and an element of a Euclidean Jordan algebra when its Peirce decomposition with respect to a Jordan frame is given. An Oppenheim type inequality is proved in this setting.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Carlson D., Markham T.L.: Schur complements of diagonally dominant matrices. Czech. Math. J. 29, 246–251 (1979)
Cohen N., De Leo S.: The quaternionic determinant. Electron. J. Linear Algebra 7, 100–111 (2000)
Cottle R.W.: On manifestations of the Schur complement. Rend. Sem. Mat. Fis. Milano 45, 31–40 (1975)
Crabtree D.E., Haynsworth E.V.: An identity for the Schur complement of a matrix. Proc. Am. Math. Soc. 22, 364–366 (1969)
Dray T., Manogue C.A.: The octonionic eigenvalue problem. Adv. Appl. Clifford Algebra 8, 341–364 (1998)
Faraut J., Korányi A.: Analysis on Symmetric Cones. Clarendon Press, Oxford (1994)
Gowda M.S., Sznajder R.: Automorphism invariance of P and GUS properties of linear transformations on Euclidean Jordan algebras. Math. Oper. Res. 31, 109–123 (2006)
Gowda M.S., Sznajder R.: Schur complements, Schur determinantal and Haynsworth inertia formulas in Euclidean Jordan algebras. Linear Algebra Appl. 432, 1553–1559 (2010)
Gowda M.S., Sznajder R., Tao J.: Some P-properties for linear transformations on Euclidean Jordan algebras. Linear Algebra Appl. 393, 203–232 (2004)
Gowda M.S., Tao J.: The Cauchy interlacing theorem in simple Euclidean Jordan algebras and some consequences. Linear Multilinear Algebra 59, 65–86 (2011)
Gowda, M.S., Tao, J.: Some inequalities involving determinants, eigenvalues, and Schur complements in Euclidean Jordan algebras, To appear in Positivity
Gowda M.S., Tao J., Moldovan M.: Some inertia theorems in Euclidean Jordan algebras. Linear Algebra Appl. 430, 1992–2011 (2009)
Horn R.A., Johnson C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)
Korányi A.: Monotone functions on formally real Jordan algebras. Math. Ann. 269, 73–76 (1984)
Loos O.: Finiteness condition in Jordan pairs. Math. Z. 206, 577–587 (1991)
Massam H., Neher E.: On transformations and determinants of Wishart variables on symmetric cones. J. Theor. Probab. 10, 867–902 (1997)
Moldovan M.M., Gowda M.S.: Strict diagonal dominance and a Gers̆gorin type theorem in Euclidean Jordan algebras. Linear Algebra Appl. 431, 148–161 (2009)
Ouellette D.V.: Schur complements and statistics. Linear Algebra Appl. 36, 187–295 (1981)
Parsons T.D.: Applications of principal pivoting, Proceedings of Princeton Symposium on Mathematical Programming, pp. 567–581. Princeton University Press, Princeton NJ (1970)
Schmieta S.H., Alizadeh F.: Extension of primal-dual interior point algorithms to symmetric cones. Math. Program., Series A 96, 409–438 (2003)
Zhang F.: Matrix Theory. Springer, New York (1999)
Zhang F.: The Schur Complement and its Applications. Springer, New York (2005)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sznajder, R., Gowda, M.S. & Moldovan, M.M. More results on Schur complements in Euclidean Jordan algebras. J Glob Optim 53, 121–134 (2012). https://doi.org/10.1007/s10898-011-9734-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-011-9734-x