Lie-Trotter Formula for the Hadamard Product
Suppose that A and B are two positive-definite matrices, then, the limit of ( A p /2 B p A p /2 ) 1/ p as p tends to 0 can be obtained by the well known Lie-Trotter formula. In this article, we generalize the usual product of matrices to the Hadamard product denoted as * which is commutative, and ob...
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| Published in | Acta mathematica scientia Vol. 40; no. 3; pp. 659 - 669 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Singapore
Springer Singapore
01.05.2020
School of Information, Beijing Wuzi University, Beijing 101149, China%College of Science, Zhengzhou University of Aeronautics, Zhengzhou 450015, China%School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China Beijing Key Laboratory on MCAACI, Beijing 100081, China |
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| Abstract | Suppose that
A
and
B
are two positive-definite matrices, then, the limit of (
A
p
/2
B
p
A
p
/2
)
1/
p
as
p
tends to 0 can be obtained by the well known Lie-Trotter formula. In this article, we generalize the usual product of matrices to the Hadamard product denoted as * which is commutative, and obtain the explicit formula of the limit (
A
p
*
B
p
)
1/
p
as
p
tends to 0. Furthermore, the existence of the limit of (
A
p
*
B
p
)
1/
p
as
p
tends to +∞ is proved. |
|---|---|
| AbstractList | Suppose that
A
and
B
are two positive-definite matrices, then, the limit of (
A
p
/2
B
p
A
p
/2
)
1/
p
as
p
tends to 0 can be obtained by the well known Lie-Trotter formula. In this article, we generalize the usual product of matrices to the Hadamard product denoted as * which is commutative, and obtain the explicit formula of the limit (
A
p
*
B
p
)
1/
p
as
p
tends to 0. Furthermore, the existence of the limit of (
A
p
*
B
p
)
1/
p
as
p
tends to +∞ is proved. Suppose that A and B are two positive-definite matrices,then,the limit of (Ap/2BpAp/2)1/p as p tends to 0 can be obtained by the well known Lie-Trotter formula.In this article,we generalize the usual product of matrices to the Hadamard product denoted as * which is commutative,and obtain the explicit formula of the limit (Ap * Bp)1/p as p tends to 0.Furthermore,the existence of the limit of (Ap * Bp)1/p as p tends to +∞ is proved. |
| Author | Wang, Jing Sun, Huafei Li, Yonggang |
| AuthorAffiliation | School of Information, Beijing Wuzi University, Beijing 101149, China%College of Science, Zhengzhou University of Aeronautics, Zhengzhou 450015, China%School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China Beijing Key Laboratory on MCAACI, Beijing 100081, China |
| AuthorAffiliation_xml | – name: School of Information, Beijing Wuzi University, Beijing 101149, China%College of Science, Zhengzhou University of Aeronautics, Zhengzhou 450015, China%School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China Beijing Key Laboratory on MCAACI, Beijing 100081, China |
| Author_xml | – sequence: 1 givenname: Jing surname: Wang fullname: Wang, Jing email: wangjingzzumath@163.com organization: School of Information, Beijing Wuzi University – sequence: 2 givenname: Yonggang surname: Li fullname: Li, Yonggang organization: College of Science, Zhengzhou University of Aeronautics – sequence: 3 givenname: Huafei surname: Sun fullname: Sun, Huafei organization: School of Mathematics and Statistics, Beijing Institute of Technology, Beijing Key Laboratory on MCAACI |
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| Cites_doi | 10.1063/1.4906367 10.1007/BF01371042 10.4153/CMB-1959-012-2 10.1007/s002080050335 10.1080/03081087408817030 10.1016/S0024-3795(99)00187-1 10.1016/S0252-9602(12)60183-0 10.1090/S0002-9939-1959-0108732-6 10.1016/0024-3795(94)90484-7 10.1007/978-1-4612-0653-8 10.1016/0024-3795(94)00014-5 10.1016/0024-3795(79)90179-4 10.1080/03081087.2015.1082957 10.1007/978-0-387-21554-9 10.1007/978-3-642-65755-9 10.1007/978-1-4757-5797-2 10.1515/crll.1911.140.1 10.1016/S0024-3795(97)00068-2 |
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| Copyright | Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences 2020 Copyright © Wanfang Data Co. Ltd. All Rights Reserved. |
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| Keywords | 15A42 Lie-Trotter formula reciprocal Lie-Trotter formula 15A16 47A56 positive-definite matrix Hadamard product |
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| References | Trotter (CR2) 1959; 10 Moalla (CR14) 2012; 32B Bhatia (CR12) 2007 Ando (CR9) 1995; 223/224 Vargas (CR17) 1983; 12 Hiai, Janas, Szafraniec, Zemánek (CR3) 1997 Johnson (CR16) 1974; 1 Ando, Zhan (CR21) 1999; 315 Zhang (CR15) 1999 Ando, Hiai (CR7) 1994; 197 Kato, Gohberg, Kac (CR1) 1978 Visick (CR10) 1998; 269 Donogue (CR22) 1974 Hall (CR13) 2003 Visick (CR20) 2000; 304 Bhatia (CR11) 1997 Marcus, Khan (CR19) 1959; 2 Marcus, Minc (CR18) 1963 Ando (CR23) 1979; 26 Audenaert, Hiai (CR6) 2016; 64 Audenaert, Datta (CR5) 2015; 56 Schur (CR8) 1911; 140 Kubo, Ando (CR4) 1980; 246 R Bhatia (305_CR12) 2007 L G Vargas (305_CR17) 1983; 12 T Ando (305_CR23) 1979; 26 G Visick (305_CR20) 2000; 304 C R Johnson (305_CR16) 1974; 1 W Donogue (305_CR22) 1974 F Kubo (305_CR4) 1980; 246 M Marcus (305_CR18) 1963 K M R Audenaert (305_CR6) 2016; 64 T Ando (305_CR9) 1995; 223/224 T Ando (305_CR21) 1999; 315 T Kato (305_CR1) 1978 R Bhatia (305_CR11) 1997 N Moalla (305_CR14) 2012; 32B M Marcus (305_CR19) 1959; 2 G Visick (305_CR10) 1998; 269 I Schur (305_CR8) 1911; 140 F Zhang (305_CR15) 1999 H Trotter (305_CR2) 1959; 10 B Hall (305_CR13) 2003 K M R Audenaert (305_CR5) 2015; 56 T Ando (305_CR7) 1994; 197 F Hiai (305_CR3) 1997 |
| References_xml | – start-page: 185 year: 1978 end-page: 195 ident: CR1 article-title: Trotter’s product formula for an arbitrary pair of self-adjoint contraction semigroups publication-title: Topics in Functional Analysis – volume: 56 start-page: 59 year: 2015 end-page: 85 ident: CR5 article-title: α-z-relative Rényi entropies publication-title: J Math Phys doi: 10.1063/1.4906367 – start-page: 119 year: 1997 end-page: 181 ident: CR3 article-title: Log-majorizations and norm inequalities for exponential operators publication-title: Linear Operators. Banach Center Publications – volume: 246 start-page: 205 year: 1980 end-page: 224 ident: CR4 article-title: Means of positive linear operators publication-title: Math Ann doi: 10.1007/BF01371042 – volume: 2 start-page: 81 year: 1959 end-page: 83 ident: CR19 article-title: A note on the Hadamard product publication-title: Canad Math Bull doi: 10.4153/CMB-1959-012-2 – volume: 315 start-page: 771 year: 1999 end-page: 780 ident: CR21 article-title: Norm inequalities related to operator monotone functions publication-title: Math Ann doi: 10.1007/s002080050335 – year: 2007 ident: CR12 publication-title: Positive Definite Matrices – volume: 1 start-page: 295 year: 1974 end-page: 307 ident: CR16 article-title: Hadamard products of matrices publication-title: Linear Multilinear A doi: 10.1080/03081087408817030 – volume: 304 start-page: 45 year: 2000 end-page: 68 ident: CR20 article-title: A quantitative version of the observation that the Hadamard product is a principal submatrix of the Kronecker product publication-title: Linear Algebra Appl doi: 10.1016/S0024-3795(99)00187-1 – volume: 32B start-page: 2329 issue: 6 year: 2012 end-page: 2340 ident: CR14 article-title: A characterization of schechter’s, essential spectrum by mean of measure of non-strictsingularity and application to matrix operator publication-title: Acta Mathematica Scientia doi: 10.1016/S0252-9602(12)60183-0 – volume: 10 start-page: 545 year: 1959 end-page: 551 ident: CR2 article-title: On the product of semigroups of operators publication-title: P Am Math Soc doi: 10.1090/S0002-9939-1959-0108732-6 – volume: 197 start-page: 113 year: 1994 end-page: 131 ident: CR7 article-title: Log majorization and complementary Golden-Thompson type inequalities publication-title: Linear Algebra Appl doi: 10.1016/0024-3795(94)90484-7 – year: 1997 ident: CR11 publication-title: Matrix Analysis doi: 10.1007/978-1-4612-0653-8 – year: 1963 ident: CR18 publication-title: A Survey of Matrix Theory and Matrix Inequalities – volume: 223/224 start-page: 57 year: 1995 end-page: 64 ident: CR9 article-title: Majorization relations for Hadamard products publication-title: Linear Algebra Appl doi: 10.1016/0024-3795(94)00014-5 – volume: 26 start-page: 203 year: 1979 end-page: 241 ident: CR23 article-title: Concavity of certain maps on positive definite matrices and applications to Hadamard products publication-title: Linear Algebra Appl doi: 10.1016/0024-3795(79)90179-4 – volume: 64 start-page: 1220 year: 2016 end-page: 1235 ident: CR6 article-title: Reciprocal Lie-Trotter formula publication-title: Linear Multilinear A doi: 10.1080/03081087.2015.1082957 – year: 2003 ident: CR13 publication-title: Lie groups, Lie algebras, and Representations: An elementary introduction doi: 10.1007/978-0-387-21554-9 – year: 1974 ident: CR22 publication-title: Monotone Matrix Functions and Analytic Continuation doi: 10.1007/978-3-642-65755-9 – year: 1999 ident: CR15 publication-title: Matrix Theory: Basic results and techniques doi: 10.1007/978-1-4757-5797-2 – volume: 140 start-page: 1 year: 1911 end-page: 28 ident: CR8 article-title: Bemerkungen zur theorie der beschränkten bilinearformen mit unendlich vielen veränderlichen publication-title: J Reine Angew Math doi: 10.1515/crll.1911.140.1 – volume: 269 start-page: 233 year: 1998 end-page: 240 ident: CR10 article-title: Majorizations of Hadamard products of matrix powers publication-title: Linear Algebra Appl doi: 10.1016/S0024-3795(97)00068-2 – volume: 12 start-page: 301 year: 1983 end-page: 320 ident: CR17 article-title: Analysis of sensitivity of reciprocal matrices publication-title: Appl Math Comput – volume-title: Positive Definite Matrices year: 2007 ident: 305_CR12 – volume: 304 start-page: 45 year: 2000 ident: 305_CR20 publication-title: Linear Algebra Appl doi: 10.1016/S0024-3795(99)00187-1 – volume: 2 start-page: 81 year: 1959 ident: 305_CR19 publication-title: Canad Math Bull doi: 10.4153/CMB-1959-012-2 – volume: 1 start-page: 295 year: 1974 ident: 305_CR16 publication-title: Linear Multilinear A doi: 10.1080/03081087408817030 – volume: 10 start-page: 545 year: 1959 ident: 305_CR2 publication-title: P Am Math Soc doi: 10.1090/S0002-9939-1959-0108732-6 – volume: 197 start-page: 113 year: 1994 ident: 305_CR7 publication-title: Linear Algebra Appl doi: 10.1016/0024-3795(94)90484-7 – volume: 32B start-page: 2329 issue: 6 year: 2012 ident: 305_CR14 publication-title: Acta Mathematica Scientia doi: 10.1016/S0252-9602(12)60183-0 – volume: 56 start-page: 59 year: 2015 ident: 305_CR5 publication-title: J Math Phys doi: 10.1063/1.4906367 – volume: 64 start-page: 1220 year: 2016 ident: 305_CR6 publication-title: Linear Multilinear A doi: 10.1080/03081087.2015.1082957 – volume-title: Monotone Matrix Functions and Analytic Continuation year: 1974 ident: 305_CR22 doi: 10.1007/978-3-642-65755-9 – volume: 140 start-page: 1 year: 1911 ident: 305_CR8 publication-title: J Reine Angew Math doi: 10.1515/crll.1911.140.1 – volume: 269 start-page: 233 year: 1998 ident: 305_CR10 publication-title: Linear Algebra Appl doi: 10.1016/S0024-3795(97)00068-2 – volume: 223/224 start-page: 57 year: 1995 ident: 305_CR9 publication-title: Linear Algebra Appl doi: 10.1016/0024-3795(94)00014-5 – volume-title: Matrix Analysis year: 1997 ident: 305_CR11 doi: 10.1007/978-1-4612-0653-8 – start-page: 119 volume-title: Linear Operators. Banach Center Publications year: 1997 ident: 305_CR3 – volume-title: Matrix Theory: Basic results and techniques year: 1999 ident: 305_CR15 doi: 10.1007/978-1-4757-5797-2 – volume-title: A Survey of Matrix Theory and Matrix Inequalities year: 1963 ident: 305_CR18 – volume: 315 start-page: 771 year: 1999 ident: 305_CR21 publication-title: Math Ann doi: 10.1007/s002080050335 – volume: 12 start-page: 301 year: 1983 ident: 305_CR17 publication-title: Appl Math Comput – volume: 246 start-page: 205 year: 1980 ident: 305_CR4 publication-title: Math Ann doi: 10.1007/BF01371042 – start-page: 185 volume-title: Topics in Functional Analysis year: 1978 ident: 305_CR1 – volume: 26 start-page: 203 year: 1979 ident: 305_CR23 publication-title: Linear Algebra Appl doi: 10.1016/0024-3795(79)90179-4 – volume-title: Lie groups, Lie algebras, and Representations: An elementary introduction year: 2003 ident: 305_CR13 doi: 10.1007/978-0-387-21554-9 |
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| Snippet | Suppose that
A
and
B
are two positive-definite matrices, then, the limit of (
A
p
/2
B
p
A
p
/2
)
1/
p
as
p
tends to 0 can be obtained by the well known... Suppose that A and B are two positive-definite matrices,then,the limit of (Ap/2BpAp/2)1/p as p tends to 0 can be obtained by the well known Lie-Trotter... |
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| Title | Lie-Trotter Formula for the Hadamard Product |
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