Skip to content
BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access February 12, 2016

Matrix rank/inertia formulas for least-squares solutions with statistical applications

  • Yongge Tian and Bo Jiang
From the journal Special Matrices


Least-Squares Solution (LSS) of a linear matrix equation and Ordinary Least-Squares Estimator (OLSE) of unknown parameters in a general linear model are two standard algebraical methods in computational mathematics and regression analysis. Assume that a symmetric quadratic matrix-valued function Φ(Z) = Q − ZPZ0 is given, where Z is taken as the LSS of the linear matrix equation AZ = B. In this paper, we establish a group of formulas for calculating maximum and minimum ranks and inertias of Φ(Z) subject to the LSS of AZ = B, and derive many quadratic matrix equalities and inequalities for LSSs from the rank and inertia formulas. This work is motivated by some inference problems on OLSEs under general linear models, while the results obtained can be applied to characterize many algebraical and statistical properties of the OLSEs.


[1] B.E. Cain, E.M. De Sá. The inertia of Hermitian matrices with a prescribed 2 × 2 block decomposition. Linear Multilinear Algebra 31(1992), 119–130. Search in Google Scholar

[2] Y. Chabrillac, J.P. Crouzeix. Definiteness and semidefiniteness of quadratic forms revisited. Linear Algebra Appl. 63(1984), 283–292. Search in Google Scholar

[3] C.M. da Fonseca. The inertia of certain Hermitian block matrices. Linear Algebra Appl. 274(1998), 193–210. 10.1016/S0024-3795(97)00338-8Search in Google Scholar

[4] J. Dancis. The possible inertias for a Hermitianmatrix and its principal submatrices. Linear Algebra Appl. 85(1987), 121–151. 10.1016/0024-3795(87)90212-6Search in Google Scholar

[5] J. Dancis. Several consequences of an inertia theorem. Linear Algebra Appl. 136(1990), 43–61. 10.1016/0024-3795(90)90020-DSearch in Google Scholar

[6] F.A. Graybill. An Introduction to Linear Statistical Models. Vol. I, McGraw–Hill, 1961. Search in Google Scholar

[7] L. Guttman. General theory and methods for matric factoring. Psychometrika 9(1944), 1–16. 10.1007/BF02288709Search in Google Scholar

[8] A. Hatcher. Algebraic Topology. Cambridge University Press, 2002. Search in Google Scholar

[9] E.V. Haynsworth. Determination of the inertia of a partitioned Hermitian matrix. Linear Algebra Appl. 1(1968), 73–81. 10.1016/0024-3795(68)90050-5Search in Google Scholar

[10] E.V. Haynsworth, A.M. Ostrowski. On the inertia of some classes of partitionedmatrices. Linear Algebra Appl. 1(1968), 299– 316. 10.1016/0024-3795(68)90009-8Search in Google Scholar

[11] I.J. James. The topology of Stiefel manifolds. Cambridge University Press, 1976. 10.1017/CBO9780511600753Search in Google Scholar

[12] C.R. Johnson, L. Rodman. Inertia possibilities for completions of partial Hermitian matrices. Linear Multilinear Algebra 16(1984), 179–195. 10.1080/03081088408817622Search in Google Scholar

[13] Y. Liu, Y. Tian. More on extremal ranks of the matrix expressions A − BX ± X*B* with statistical applications. Numer. Linear Algebra Appl. 15(2008), 307–325. Search in Google Scholar

[14] Y. Liu, Y. Tian. Max-min problems on the ranks and inertias of the matrix expressions A − BXC ± (BXC)* with applications. J. Optim. Theory Appl. 148(2011), 593–622. Search in Google Scholar

[15] G. Marsaglia, G.P.H. Styan. Equalities and inequalities for ranks of matrices. Linear Multilinear Algebra 2(1974), 269–292. 10.1080/03081087408817070Search in Google Scholar

[16] S.K. Mitra. The matrix equations AX = C, XB = D. Linear Algebra Appl. 59(1984), 171–181. 10.1016/0024-3795(84)90166-6Search in Google Scholar

[17] R. Penrose. A generalized inverse for matrices. Proc. Cambridge Phil. Soc. 51(1955), 406–413. 10.1017/S0305004100030401Search in Google Scholar

[18] S. Puntanen, G.P.H. Styan, J. Isotalo. Matrix Tricks for Linear Statistical Models, Our Personal Top Twenty. Springer, 2011. 10.1007/978-3-642-10473-2Search in Google Scholar

[19] S.R. Searle. Linear Models. Wiley, 1971. Search in Google Scholar

[20] Y. Tian. Equalities and inequalities for inertias of Hermitian matrices with applications. Linear Algebra Appl. 433(2010), 263–296. 10.1016/j.laa.2010.02.018Search in Google Scholar

[21] Y. Tian. Maximization and minimization of the rank and inertia of the Hermitian matrix expression A − BX − (BX)* with applications. Linear Algebra Appl. 434(2011), 2109–2139. 10.1016/j.laa.2010.12.010Search in Google Scholar

[22] Y. Tian. Solving optimization problems on ranks and inertias of some constrained nonlinearmatrix functions via an algebraic linearization method. Nonlinear Analysis 75(2012), 717–734. 10.1016/ in Google Scholar

[23] Y. Tian. Formulas for calculating the extremum ranks and inertias of a four-term quadratic matrix-valued function and their applications. Linear Algebra Appl. 437(2012), 835–859. 10.1016/j.laa.2012.03.021Search in Google Scholar

[24] Y. Tian. Solutions to 18 constrained optimization problems on the rank and inertia of the linear matrix function A + BXB*. Math. Comput. Modelling 55(2012), 955–968. 10.1016/j.mcm.2011.09.022Search in Google Scholar

[25] Y. Tian. Some optimization problems on ranks and inertias of matrix-valued functions subject to linear matrix equation restrictions. Banach J. Math. Anal. 8(2014), 148–178. 10.15352/bjma/1381782094Search in Google Scholar

[26] Y. Tian. A new derivation of BLUPs under a random-effects model. Metrika 78(2015), 905–918. 10.1007/s00184-015-0533-0Search in Google Scholar

[27] Y. Tian. A matrix handling of predictions under a general linear random-effects model with new observations. Electron. J. Linear Algebra 29(2015), 30–45. 10.13001/1081-3810.2895Search in Google Scholar

[28] Y. Tian. A survey on rank and inertia optimization problems of the matrix-valued function A + BXB*. Numer. Algebra Contr. Optim. 5(2015), 289–326. 10.3934/naco.2015.5.289Search in Google Scholar

[29] Y. Tian. How to characterize properties of general Hermitian quadratic matrix-valued functions by rank and inertia. In: Advances in Linear Algebra Researches, I. Kyrchei, (ed.), Nova Publishers, New York, pp. 150–183, 2015. Search in Google Scholar

[30] Y. Tian. Some equalities and inequalities for covariance matrices of estimators under linear model. Statist. Papers, DOI:10.1007/s00362-015-0707-x. 10.1007/s00362-015-0707-xSearch in Google Scholar

[31] Y. Tian,W. Guo. On comparison of dispersionmatrices of estimators under a constrained linear model. Stat. Methods Appl., DOI:10.1007/s10260-016-0350-2. 10.1007/s10260-016-0350-2Search in Google Scholar

Received: 2015-6-11
Accepted: 2016-2-1
Published Online: 2016-2-12

©2016 Yongge Tian and Bo Jiang

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Downloaded on 23.2.2024 from
Scroll to top button