PL EN

Widoczny [Schowaj] Abstrakt
Liczba wyników
Czasopismo

## Open Mathematics

2012 | 10 | 1 | 329-351
Tytuł artykułu

### Inertias and ranks of some Hermitian matrix functions with applications

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let S be a given set consisting of some Hermitian matrices with the same size. We say that a matrix A ∈ S is maximal if A − W is positive semidefinite for every matrix W ∈ S. In this paper, we consider the maximal and minimal inertias and ranks of the Hermitian matrix function f(X,Y) = P − QXQ* − TYT*, where * means the conjugate and transpose of a matrix, P = P*, Q, T are known matrices and for X and Y Hermitian solutions to the consistent matrix equations AX =B and YC = D respectively. As applications, we derive the necessary and sufficient conditions for the existence of maximal matrices of $$H = \{ f(X,Y) = P - QXQ* - TYT* : AX = B,YC = D,X = X*, Y = Y*\} .$$ The corresponding expressions of the maximal matrices of H are presented when the existence conditions are met. In this case, we further prove the matrix function f(X,Y)is invariant under changing the pair (X,Y). Moreover, we establish necessary and sufficient conditions for the system of matrix equations $$AX = B, YC = D, QXQ* + TYT* = P$$ to have a Hermitian solution and the system of matrix equations $$AX = C, BXB* = D$$ to have a bisymmetric solution. The explicit expressions of such solutions to the systems mentioned above are also provided. In addition, we discuss the range of inertias of the matrix functions P ± QXQ* ± TYT* where X and Y are a nonnegative definite pair of solutions to some consistent matrix equations. The findings of this pape extend some known results in the literature.
Słowa kluczowe
EN
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
Numer
Strony
329-351
Opis fizyczny
Daty
wydano
2012-02-01
online
2011-12-09
Twórcy
autor
autor
• Shanghai University
autor
• Shanghai University
Bibliografia
•  Bittanti S., Bolzern P., Colaneri P., Inertia theorems for Lyapunov and Riccati equations - an updated view, In: Linear Algebra in Signals, Systems, and Control, Boston, 1986, SIAM, Philadelphia, 1988, 11–35
•  Canto e Castro L., Dias S., da Graça Temido M., Looking for max-semistability: A new test for the extreme value condition, J. Statist. Plann. Inference, 2011, 141(9), 3005–3020 http://dx.doi.org/10.1016/j.jspi.2011.03.020
•  Cantoni A., Butler P., Eigenvalues and eigenvectors of symmetric centrosymmetric matrices, Linear Algebra Appl., 1976, 13(3), 275–288 http://dx.doi.org/10.1016/0024-3795(76)90101-4
•  Chang X.W., Wang J.S., The symmetric solution of the matrix equations AX + YA = C, AXA T + BYB T = C, and (A TXA;B TXB) = (C,D), Linear Algebra Appl., 1993, 179, 171–189 http://dx.doi.org/10.1016/0024-3795(93)90328-L
•  Chu D.L., Chan H.C., Ho D.W.C., Regularization of singular systems by derivative and proportional output feedback, SIAM J. Matrix Anal. Appl., 1998, 19(1), 21–38 http://dx.doi.org/10.1137/S0895479895270963
•  Chu D., De Lathauwer L., De Moor B., On the computation of the restricted singular value decomposition via the cosine-sine decomposition, SIAM J. Matrix Anal. Appl., 2000, 22(2), 580–601 http://dx.doi.org/10.1137/S0895479898346983
•  Chu D., Hung Y.S., Woerdeman H.J., Inertia and rank characterizations of some matrix expressions, SIAM J. Matrix Anal. Appl., 2009, 31(3), 1187–1226 http://dx.doi.org/10.1137/080712945
•  Datta B.N., Stability and inertia, Linear Algebra Appl., 1999, 302/303, 563–600 http://dx.doi.org/10.1016/S0024-3795(99)00213-X
•  Dehghan M., Hajarian M., An iterative method for solving the generalized coupled Sylvester matrix equations over generalized bisymmetric matrices, Appl. Math. Model., 2010, 34(3), 639–654 http://dx.doi.org/10.1016/j.apm.2009.06.018
•  Dehghan M., Hajarian M., The general coupled matrix equations over generalized bisymmetric matrices, Linear Algebra Appl., 2010, 432(6), 1531–1552 http://dx.doi.org/10.1016/j.laa.2009.11.014
•  Higham N.J., Cheng S.H., Modifying the inertia of matrices arising in optimization, Linear Algebra Appl., 1998, 275/276, 261–279 http://dx.doi.org/10.1016/S0024-3795(97)10015-5
•  Khan I.A., Wang Q.-W., Song G.-J., Minimal ranks of some quaternion matrix expressions with applications, Appl. Math. Comput., 2010, 217(5), 2031–2040 http://dx.doi.org/10.1016/j.amc.2010.07.004
•  Khatri C.G., Mitra S.K., Hermitian and nonnegative definite solutions of linear matrix equations, SIAM J. Appl. Math., 1976, 31(4), 579–585 http://dx.doi.org/10.1137/0131050
•  Marsaglia G., Styan G.P.H., Equalities and inequalities for ranks of matrices, Linear and Multilinear Algebra, 1974, 2, 269–292 http://dx.doi.org/10.1080/03081087408817070
•  Peng Z., Hu X., Zhang L., The inverse problem of bisymmetric matrices with a submatrix constraint, Numer. Linear Algebra Appl., 2004, 11(1), 59–73 http://dx.doi.org/10.1002/nla.333
•  Peng Z., Hu X., Zhang L., The bisymmetric solutions of the matrix equation A 1X 1B 1 + A 2X 2B 2 + ... + A l X l B l = C and its optimal approximation, Linear Algebra Appl., 2007, 426, 583–595 http://dx.doi.org/10.1016/j.laa.2007.05.034
•  Pereira E.S., On solvents of matrix polynomials, Appl. Numer. Math., 2003, 47(2), 197–208 http://dx.doi.org/10.1016/S0168-9274(03)00058-8
•  Pressman I.S., Matrices with multiple symmetry properties: applications of centro-Hermitian and per-Hermitian matrices, Linear Algebra Appl., 1998, 284(1–3), 239–258 http://dx.doi.org/10.1016/S0024-3795(98)10144-1
•  Reid R.M., Some eigenvalue properties of persymmetric matrices, SIAM Rev., 1997, 39(2), 313–316 http://dx.doi.org/10.1137/S0036144595294801
•  Sayed A.H., Hassibi B., Kailath T., Inertia properties of indefinite quadratic forms, IEEE Signal Processing Letters, 1996, 3(2), 57–59 http://dx.doi.org/10.1109/97.484217
•  Stykel T., Stability and inertia theorems for generalized Lyapunov equations, Linear Algebra Appl., 2002, 355, 297–314 http://dx.doi.org/10.1016/S0024-3795(02)00354-3
•  Tian Y., Upper and lower bounds for ranks of matrix expressions using generalized inverses, Linear Algebra Appl., 2002, 355, 187–214 http://dx.doi.org/10.1016/S0024-3795(02)00345-2
•  Tian Y., Equalities and inequalities for inertias of Hermitian matrices with applications, Linear Algebra Appl., 2010, 433(2), 263–296 http://dx.doi.org/10.1016/j.laa.2010.02.018
•  Wang Q.-W., Bisymmetric and centrosymmetric solutions to systems of real quaternion matrix equations, Comput. Math. Appl., 2005, 49(5–6), 641–650 http://dx.doi.org/10.1016/j.camwa.2005.01.014
•  Wang Q., Chang H., Lin C., On the centro-symmetric solution of a system of matrix equations over a regular ring with identity, Algebra Colloq., 2007, 14(4), 555–570
•  Wang Q.-W., Fei-Zhang, The reflexive re-nonnegative definite solution to a quaternion matrix equation, Electron. J. Linear Algebra, 2008, 17, 88–101
•  Wang Q.W., Jiang J., Extreme ranks of (skew-)Hermitian solutions to a quaternion matrix equation, Electron. J. Linear Algebra, 2010, 20, 552–573
•  Wang Q.-W., Li C.-K., Ranks and the least-norm of the general solution to a system of quaternion matrix equations, Linear Algebra Appl., 2009, 430(5–6), 1626–1640 http://dx.doi.org/10.1016/j.laa.2008.05.031
•  Wang Q.-W., Song G.-J., Lin C.-Y., Extreme ranks of the solution to a consistent system of linear quaternion matrix equations with an application, Appl. Math. Comput., 2007, 189(2), 1517–1532 http://dx.doi.org/10.1016/j.amc.2006.12.039
•  Wang Q.-W., Song G.-J., Lin C.-Y., Rank equalities related to the generalized inverse A T,S(2) with applications, Appl. Math. Comput., 2008, 205(1), 370–382 http://dx.doi.org/10.1016/j.amc.2008.08.016
•  Wang Q., Song G., Liu X., Maximal and minimal ranks of the common solution of some linear matrix equations over an arbitrary division ring with applications, Algebra Colloq., 2009, 16(2), 293–308
•  Wang Q.-W., Sun J.-H., Li S.-Z., Consistency for bi(skew)symmetric solutions to systems of generalized Sylvester equations over a finite central algebra, Linear Algebra Appl., 2002, 353, 169–182 http://dx.doi.org/10.1016/S0024-3795(02)00303-8
•  Wang Q.W., van der Woude J.W., Yu S.W., An equivalence canonical form of a matrix triplet over an arbitrary division ring with applications, Sci. China Math., 2011, 54(5), 907–924 http://dx.doi.org/10.1007/s11425-010-4154-9
•  Wang Q.-W., Wu Z.-C., Common Hermitian solutions to some operator equations on Hilbert C*-modules, Linear Algebra Appl., 2010, 432(12), 3159–3171 http://dx.doi.org/10.1016/j.laa.2010.01.015
•  Wang Q., Yu S., Xie W., Extreme ranks of real matrices in solution of the quaternion matrix equation AXB = C with applications, Algebra Colloq., 2010, 17(2), 345–360
•  Wang Q.-W., Yu S.-W., Zhang Q., The real solution to a system of quaternion matrix equations with applications, Comm. Algebra, 2009, 37(6), 2060–2079 http://dx.doi.org/10.1080/00927870802317590
•  Wang Q.W., Zhang H.-S., Song G.-J., A new solvable condition for a pair of generalized Sylvester equations, Electron. J. Linear Algebra, 2009, 18, 289–301
•  Wang Q.-W., Zhang H.-S., Yu S.-W., On solutions to the quaternion matrix equation AXB + CYD = E, Electron. J. Linear Algebra, 2008, 17, 343–358
•  Wimmer H.K., Inertia theorems for matrices, controllability, and linear vibrations, Linear Algebra Appl., 1974, 8, 337–343 http://dx.doi.org/10.1016/0024-3795(74)90060-3
•  Xie D., Hu X., Sheng Y., The solvability conditions for the inverse eigenproblems of symmetric and generalized centro-symmetric matrices and their approximations, Linear Algebra Appl., 2006, 418(1), 142–152 http://dx.doi.org/10.1016/j.laa.2006.01.027
•  Xie D., Zhang L., Hu X., The solvability conditions for the inverse problem of bisymmetric nonnegative definite matrices, J. Comput. Math., 2000, 18(6), 597–608
•  Xie D., Zhang Z., Liu Z., Theory and method for updating least-squares finite element model of symmetric generalized centro-symmetric matrices, J. Comput. Appl. Math., 2008, 216(2), 484–497 http://dx.doi.org/10.1016/j.cam.2007.05.030
•  Yuan S., Liao A., Lei Y., Least squares Hermitian solution of the matrix equation (AXB, CXD) = (E, F) with the least norm over the skew field of quaternions, Math. Comput. Modelling, 2008, 48(1–2), 91–100 http://dx.doi.org/10.1016/j.mcm.2007.08.009
•  Yuan S., Liao A., Lei Y., Inverse eigenvalue problems of tridiagonal symmetric matrices and tridiagonal bisymmetric matrices, Comput. Math. Appl., 2008, 55(11), 2521–2532 http://dx.doi.org/10.1016/j.camwa.2007.10.006
•  Zhang H.-S., Wang Q.-W., Ranks of submatrices in a general solution to a quaternion system with applications, Bull. Korean Math. Soc., 2011, 48(5), 969–990 http://dx.doi.org/10.4134/BKMS.2011.48.5.969
•  Zhang Q., Wang Q.-W., The (P,Q)-(skew)symmetric extremal rank solutions to a system of quaternion matrix equations, Appl. Math. Comput., 2011, 217(22), 9286–9296 http://dx.doi.org/10.1016/j.amc.2011.04.011
•  Zhang X., The general Hermitian nonnegative-definite solution to the matrix equation AXA* + BYB* = C, Acta Math. Acad. Paedagog. Nyhazi. (N.S.), 2005, 21(1), 33–42
•  Zhao L., Hu X., Zhang L., Least squares solutions to AX = B for bisymmetric matrices under a central principal submatrix constraint and the optimal approximation, Linear Algebra Appl., 2008, 428(4), 871–880 http://dx.doi.org/10.1016/j.laa.2007.08.019
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-011-0117-9 JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.