A partial ordering approach to characterize properties of a pair of orthogonal projectors
It is known that within the set of orthogonal projectors (Hermitian idempotent matrices) certain matrix partial orderings coincide in the sense that when two orthogonal projectors are ordered with respect to one of the orderings, then they are also ordered with respect to the others. This concerns,...
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| Published in | Indian journal of pure and applied mathematics Vol. 52; no. 2; pp. 323 - 334 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
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Indian National Science Academy
01.06.2021
Springer Nature B.V |
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| Abstract | It is known that within the set of orthogonal projectors (Hermitian idempotent matrices) certain matrix partial orderings coincide in the sense that when two orthogonal projectors are ordered with respect to one of the orderings, then they are also ordered with respect to the others. This concerns, inter alia, the star, minus, diamond, sharp, core, and Löwner orderings. The situation changes, though, when instead of two orthogonal projectors, various functions of the pair (being either no longer Hermitian or no longer idempotent) are compared. The present paper provides an extensive investigation of the matrix partial orderings of functions of two orthogonal projectors. In addition to the six orderings mentioned above, three further binary functions are covered by the analysis, one of which is the space preordering. A particular attention is paid to the requirements that either product, sum, or difference of two orthogonal projectors is itself an orthogonal projector, i.e., inherits both features, Hermitianness and idempotency. Links of the results obtained with the research areas of applied origin (e.g., physics and statistics) are pointed out as well. |
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| AbstractList | It is known that within the set of orthogonal projectors (Hermitian idempotent matrices) certain matrix partial orderings coincide in the sense that when two orthogonal projectors are ordered with respect to one of the orderings, then they are also ordered with respect to the others. This concerns, inter alia, the star, minus, diamond, sharp, core, and Löwner orderings. The situation changes, though, when instead of two orthogonal projectors, various functions of the pair (being either no longer Hermitian or no longer idempotent) are compared. The present paper provides an extensive investigation of the matrix partial orderings of functions of two orthogonal projectors. In addition to the six orderings mentioned above, three further binary functions are covered by the analysis, one of which is the space preordering. A particular attention is paid to the requirements that either product, sum, or difference of two orthogonal projectors is itself an orthogonal projector, i.e., inherits both features, Hermitianness and idempotency. Links of the results obtained with the research areas of applied origin (e.g., physics and statistics) are pointed out as well. |
| Author | Baksalary, Oskar Maria Trenkler, Götz |
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Martin, A partial order on classical and quantum states, New Structures for Physics (B. Coecke, ed.), Springer-Verlag, Berlin, 2011, 593-683. – reference: BaksalaryJKA relationship between the star and minus orderingsLinear Algebra Appl.19868216316785896910.1016/0024-3795(86)90149-7 – reference: DrazinMPNatural structures on semigroups with involutionBull. Amer. Math. Soc.19788413914148623410.1090/S0002-9904-1978-14442-5 – reference: MarsagliaGStyanGPHEqualities and inequalities for ranks of matricesLinear Multilinear Algebra1974226929238484010.1080/03081087408817070 – reference: MostajeranCSepulchreROrdering positive definite matricesInfo. Geo.20181287313397467210.1007/s41884-018-0003-7 – reference: HarvilleDAMatrix Algebra From a Statistician’s Perspective1997New YorkSpringer10.1007/b98818 – reference: BaksalaryOMTrenklerGOn subspace distances determined by the Frobenius normLinear Algebra Appl.2014448245263318298310.1016/j.laa.2014.01.017 – reference: BaksalaryOMTrenklerGColumn space equalities for orthogonal projectorsAppl. Math. Comput.200921251952925319781169.15007 – reference: PuntanenSStyanGPHIsotaloJMatrix Tricks for Linear Statistical Models - Our Personal Top Twenty2011BerlinSpringer-Verlag10.1007/978-3-642-10473-2 – reference: BaksalaryOMTrenklerGCore inverse of matricesLinear Multilinear Algebra201058681697272275210.1080/03081080902778222 – reference: BaksalaryJKMitraSKLeft-star and right-star partial orderingsLinear Algebra Appl.19911497389109287010.1016/0024-3795(91)90326-R – reference: MitraSKBhimasankaramPMalikSBMatrix Partial Orders2010World Scientific, SingaporeShorted Operators and Applications1203.15023 – reference: M. Fattore and R. Bruggemann, ed., Partial Order Concepts in Applied Sciences, Springer, Cham, 2017. – reference: BaksalaryJKHaukeJA further algebraic version of Cochran’s theorem and matrix partial orderingsLinear Algebra Appl.1999127157169104880010.1016/0024-3795(90)90341-9 – reference: BrittinWESakakuraAYProjection operator techniques in physicsJ. Math. 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Edinburgh Math. Soc. doi: 10.1017/S0013091500003801 – volume-title: Matrix Tricks for Linear Statistical Models - Our Personal Top Twenty year: 2011 ident: 138_CR28 doi: 10.1007/978-3-642-10473-2 – volume-title: Matrix Algebra From a Statistician’s Perspective year: 1997 ident: 138_CR18 doi: 10.1007/b98818 – volume: 82 start-page: 163 year: 1986 ident: 138_CR2 publication-title: Linear Algebra Appl. doi: 10.1016/0024-3795(86)90149-7 – ident: 138_CR10 doi: 10.1016/0024-3795(79)90158-7 – volume: 62 start-page: 1629 year: 2014 ident: 138_CR20 publication-title: Linear Multilinear Algebra doi: 10.1080/03081087.2013.839676 – volume: 92 start-page: 17 year: 1987 ident: 138_CR23 publication-title: Linear Algebra Appl. doi: 10.1016/0024-3795(87)90248-5 – ident: 138_CR1 – volume: 212 start-page: 519 year: 2009 ident: 138_CR6 publication-title: Appl. Math. Comput. doi: 10.1016/j.amc.2009.02.042 – volume: 1 start-page: 287 year: 2018 ident: 138_CR25 publication-title: Info. 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Phys. doi: 10.1063/1.526284 – volume-title: Partial Ordering Methods in Nonlinear Problems year: 2004 ident: 138_CR14 – volume-title: Generalized Inverse of Matrices and Its Applications year: 1971 ident: 138_CR29 – volume: 127 start-page: 157 year: 1999 ident: 138_CR4 publication-title: Linear Algebra Appl. doi: 10.1016/0024-3795(90)90341-9 – ident: 138_CR11 doi: 10.1007/978-3-642-12821-9_10 – ident: 138_CR17 – volume: 25 start-page: 1 year: 1980 ident: 138_CR15 publication-title: Math. Japon. |
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| Title | A partial ordering approach to characterize properties of a pair of orthogonal projectors |
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