zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
The (P,Q)-(skew)symmetric extremal rank solutions to a system of quaternion matrix equations. (English) Zbl 1222.15022

The paper deals with extremal rank solutions to a system of quaternion matrix equations.

H m×n denotes the set of m×n matrices over the real quaternion algebra

H={a 0 +a 1 i+a 2 j+a 3 k:i 2 =j 2 =k 2 =ijk=-1anda 0 ,a 1 ,a 2 ,a 3 arerealnumbers}·

A matrix AH m×n is called (P,Q)-symmetric (or (P,Q)-skewsymmetric) if A=PAQ (or A=-PAQ), where PH m×m and QH n×n are involution matrices. Consider the system of matrix equations over H

AX=B,XC=D·(*)

In this work, the authors analyze the (P,Q)-(skew)symmetric maximal and minimal rank solutions of this system. They obtain necessary and sufficient conditions for the existence of (P,Q)-symmetric and (P,Q)-skewsymmetric solutions to the above system and give the expressions of such solutions when the solvability conditions are satisfied.

The authors also establish formulas of maximal and minimal ranks of (P,Q)-symmetric and (P,Q)-skewsymmetric solutions of (*) and derive the expressions of (P,Q)-(skew)symmetric maximal and minimal rank solutions of (*).

Finally, the authors present a numerical example that confirms the theoretical results obtained.

MSC:
15A24Matrix equations and identities
15A33Matrices over special rings
15A09Matrix inversion, generalized inverses
References:
[1]Hungerford, T. W.: Algebra, (1980)
[2]Zhang, F.: Quaternions and matrices of quaternions, Linear algebra appl. 251, 21-57 (1997) · Zbl 0873.15008 · doi:10.1016/0024-3795(95)00543-9
[3]De Leo, S.; Scolarici, G.: Right eigenvalue equation in quaternionic quantum mechanics, J. phys. A 33, 2971-2995 (2000) · Zbl 0954.81008 · doi:10.1088/0305-4470/33/15/306
[4]Sangwine, S. J.; Ell, T. A.; Moxey, C. E.: Vector phase correlation, Electron. lett. 37, No. 25, 1513-1515 (2001)
[5]Farenick, D. R.; Pidkowich, B. A. F.: The spectral theorem in quaternions, Linear algebra appl. 371, 75-102 (2003) · Zbl 1030.15015 · doi:10.1016/S0024-3795(03)00420-8
[6]Moxey, C. E.; Sangwine, S. J.; Ell, T. A.: Hypercomplex correlation techniques for vector images, IEEE trans. Signal process. 51, No. 7, 1177-1199 (2003)
[7]N. LE Bihan, S.J. Sangwine, Color image decomposition using quaternion singular value decomposition, in: IEEE International Conference on Visual Information Engineering (VIE), Guildford, UK, 2003.
[8]Le Bihan, N.; Mars, J.: Singular value decomposition of matrices of quaternions: a new tool for vector-sensor signal processing, Signal process. 84, No. 7, 1177-1199 (2004) · Zbl 1154.94331 · doi:10.1016/j.sigpro.2004.04.001
[9]Sangwine, S. J.; Bihan, N. L.: Quaternion singular value decomposition based on bidiagonalization to a real or complex matrix using quaternion householder transformations, Appl. math. Comput. 182, No. 1, 727-738 (2006) · Zbl 1109.65037 · doi:10.1016/j.amc.2006.04.032
[10]Chen, H. C.; Sameh, A.: Numerical linear algebra algorithms on the cedar system, Parallel computations and their impact on mechanics 86, 101-125 (1987)
[11]H.C. Chen, The SAS Domain Decomposition Method for Structural Analysis, CSRD Tech. Report 754, Center for Supercomputing Research and Development, University of Illinois, Urbana, IL, 1988.
[12]Chen, H. C.: Generalized reflexive matrices: special properties and application, SIAM J. Matrix anal. Appl. 19, 140-153 (1998) · Zbl 0910.15005 · doi:10.1137/S0895479895288759
[13]Trench, W. F.: Minimization problems for (R,S)-symmetric and (R,S)-skew symmetric matrices, Linear algebra appl. 389, 23-31 (2004) · Zbl 1059.15019 · doi:10.1016/j.laa.2004.03.035
[14]Mitra, S. K.: Fixed rank solutions of linear matrix equations, Sankhya ser. A 35, 387-392 (1972) · Zbl 0261.15008
[15]Mitra, S. K.: The matrix equations AX=C, XB=D, Linear algebra appl. 59, 171-181 (1984) · Zbl 0543.15011 · doi:10.1016/0024-3795(84)90166-6
[16]Baksalary, J. K.: Nonnegative definite and positive definite solutions to the matrix equation AXA*=B, Linear multilinear algebra 16, 133-139 (1984) · Zbl 0552.15009 · doi:10.1080/03081088408817616
[17]Uhlig, F.: On the matrix equation AX=B with applications to the generators of controllability matrix, Linear algebra appl. 85, 203-209 (1987) · Zbl 0612.15006 · doi:10.1016/0024-3795(87)90217-5
[18]Mitra, S. K.: A pair of simultaneous linear matrix equations A1XB1=C1, A2XB2=C2 and a matrix programming problem, Linear algebra appl. 131, 107-123 (1990) · Zbl 0712.15010 · doi:10.1016/0024-3795(90)90377-O
[19]Chu, D.; Chan, H. C.; Ho, D. W. C.: Regularization of singular systems by derivative and proportional output feedback, SIAM J. Matrix anal. Appl. 19, 21-38 (1998) · Zbl 0912.93027 · doi:10.1137/S0895479895270963
[20]Chu, D.; Mehrmann, V.; Nichols, N. K.: Minimum norm regularization of descriptor systems by mixed output feedback, Linear algebra appl. 296, 39-77 (1999) · Zbl 0959.93032 · doi:10.1016/S0024-3795(99)00108-1
[21]Puntanen, S.; Styan, G. P. H.: Two matrix-based proofs that the linear estimator gy is the best linear unbiased estimator, J. statist. Plan. inference 88, 173-179 (2000) · Zbl 0964.62054 · doi:10.1016/S0378-3758(00)00076-8
[22]Qian, H.; Tian, Y.: Partially superfluous observations, Econ. theory 22, 529-536 (2006) · Zbl 1125.62069 · doi:10.1017/S0266466606060269
[23]Tian, Y.; Wiens, D. P.: On equality and proportionality of ordinary least squares, weighted least squares and best linear unbiased estimators in the general linear model, Statist. probab. Lett. 76, 1265-1272 (2006) · Zbl 1123.62038 · doi:10.1016/j.spl.2006.01.005
[24]Wang, Q. W.: Bisymmetric and centrosymmetric solutions to systems of real quaternion matrix equations, Comput. math. Appl. 49/5-6, 641-650 (2005) · Zbl 1138.15003 · doi:10.1016/j.camwa.2005.01.014
[25]Wang, Q. W.: The general solution to a system of real quaternion matrix equations, Comput. math. Appl. 49/5-6, 665-675 (2005) · Zbl 1138.15004 · doi:10.1016/j.camwa.2004.12.002
[26]Qiu, Y.; Zhang, Z.; Lu, J.: The matrix equations AX=B, CX=D with PX=sXP constraint, Appl. math. Comput. 189, 1428-1434 (2007) · Zbl 1124.15009 · doi:10.1016/j.amc.2006.12.046
[27]Li, F.; Hu, X.; Zhang, L.: The generalized reflexive solution for a class of matrix equations (AX=B XC=D), Acta math. Sci. 28B, No. 1, 185-193 (2008) · Zbl 1150.15006 · doi:10.1016/S0252-9602(08)60019-3
[28]Xiao, Q. F.; Hu, X. Y.; Zhang, L.: The symmetric minimal rank solution of the matrix equation AX=B and the optimal approximation, Electron. J. Linear algebra 18, 264-271 (2009) · Zbl 1171.65387 · doi:emis:journals/ELA/ela-articles/abstracts/abs_vol18_pp264-273.pdf
[29]Peng, Z. -Y.; Hu, X. Y.: The reflexive and anti-reflexive solutions of matrix equation AX=B, Linear algebra appl. 375, 147-155 (2003) · Zbl 1050.15016 · doi:10.1016/S0024-3795(03)00607-4
[30]Trench, W. F.: Inverse eigenproblems and associated approximation problems for matrices with generalized symmetry or skew symmetry, Linear algebra appl. 380, 199-211 (2004) · Zbl 1087.15013 · doi:10.1016/j.laa.2003.10.007
[31]Wang, Q. W.; Chang, H. X.; Lin, C. Y.: P-(skew)symmetric common solutions to a pair of quaternion matrix equations, Appl. math. Comput. 195, 721-732 (2008) · Zbl 1149.15011 · doi:10.1016/j.amc.2007.05.021
[32]Wang, Q. W.; Wu, Z. C.; Lin, C. Y.: Extremal ranks of a quaternion matrix expression subject to consistent systems of quaternion matrix equations with applications, Appl. math. Comput. 182, No. 2, 1755-1764 (2006) · Zbl 1108.15014 · doi:10.1016/j.amc.2006.06.012
[33]Tian, Y.; Cheng, S.: The maximal and minimal ranks of A - BXC with applications, New York J. Math. 9, 345-362 (2003) · Zbl 1036.15004 · doi:emis:journals/NYJM/j/2003/9-18nf.htm