2012, 2(2): 293-299. doi: 10.3934/naco.2012.2.293

How to transform matrices $U_1, \ldots, U_p$ to matrices $V_1, \ldots, V_p$ so that $V_i V_j^T= {\mathbb O} $ if $ i \neq j $?

1. 

School of Computer Science, Engineering and Mathematics, Flinders University, Bedford Park, SA 5042, Australia

2. 

School of Mathematics and Statistics, University of South Australia, Mawson Lakes, SA 5095, Australia

Received  September 2011 Revised  April 2012 Published  May 2012

Methods of a transformation of matrices $U_1, \ldots, U_p$ to matrices $V_1, \ldots, V_p$ are proposed so that $V_i V_j^T={\mathbb O} $ for $ i\neq j $ and $i,j =1,\ldots,p $. We consider unconstrained and constrained problems associated with such a transformation. Solutions of the both problems are provided.
Citation: Vladimir Ejov, Anatoli Torokhti. How to transform matrices $U_1, \ldots, U_p$ to matrices $V_1, \ldots, V_p$ so that $V_i V_j^T= {\mathbb O} $ if $ i \neq j $?. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 293-299. doi: 10.3934/naco.2012.2.293
References:
[1]

T. L. Boullion and P. L. Odell, "Generized Inverse Matrices," John Willey & Sons, Inc., New York, 1972. Google Scholar

[2]

S. Friedland, A. Niknejad and L. Chihara, A simultaneous reconstruction of missing data in DNA microarrays, Linear Alg. Appl., 416 (2006), 8-28. doi: 10.1016/j.laa.2005.05.009.  Google Scholar

[3]

J. Listgarten, C. Kadie, E. Schadt and D. Heckerman, Correction for hidden confounders in the genetic analysis of gene expression, Proc. Natl. Acad. Sci. USA, 107 (2010), 16465-16470. doi: 10.1073/pnas.1002425107.  Google Scholar

[4]

A. Torokhti and P. Howlett, "Computational Methods for Modelling of Nonlinear Systems," Elsevier, 2007.  Google Scholar

[5]

A. Torokhti and P. Howlett, Optimal transform formed by a combination of nonlinear operators: The case of data dimensionality reduction, IEEE Trans. on Signal Processing, 54 (2006), 1431-1444. doi: 10.1109/TSP.2006.870560.  Google Scholar

show all references

References:
[1]

T. L. Boullion and P. L. Odell, "Generized Inverse Matrices," John Willey & Sons, Inc., New York, 1972. Google Scholar

[2]

S. Friedland, A. Niknejad and L. Chihara, A simultaneous reconstruction of missing data in DNA microarrays, Linear Alg. Appl., 416 (2006), 8-28. doi: 10.1016/j.laa.2005.05.009.  Google Scholar

[3]

J. Listgarten, C. Kadie, E. Schadt and D. Heckerman, Correction for hidden confounders in the genetic analysis of gene expression, Proc. Natl. Acad. Sci. USA, 107 (2010), 16465-16470. doi: 10.1073/pnas.1002425107.  Google Scholar

[4]

A. Torokhti and P. Howlett, "Computational Methods for Modelling of Nonlinear Systems," Elsevier, 2007.  Google Scholar

[5]

A. Torokhti and P. Howlett, Optimal transform formed by a combination of nonlinear operators: The case of data dimensionality reduction, IEEE Trans. on Signal Processing, 54 (2006), 1431-1444. doi: 10.1109/TSP.2006.870560.  Google Scholar

[1]

Zheng-Jian Bai, Xiao-Qing Jin, Seak-Weng Vong. On some inverse singular value problems with Toeplitz-related structure. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 187-192. doi: 10.3934/naco.2012.2.187

[2]

Fang Zeng, Pablo Suarez, Jiguang Sun. A decomposition method for an interior inverse scattering problem. Inverse Problems & Imaging, 2013, 7 (1) : 291-303. doi: 10.3934/ipi.2013.7.291

[3]

Daijun Jiang, Hui Feng, Jun Zou. Overlapping domain decomposition methods for linear inverse problems. Inverse Problems & Imaging, 2015, 9 (1) : 163-188. doi: 10.3934/ipi.2015.9.163

[4]

Tao Wu, Yu Lei, Jiao Shi, Maoguo Gong. An evolutionary multiobjective method for low-rank and sparse matrix decomposition. Big Data & Information Analytics, 2017, 2 (1) : 23-37. doi: 10.3934/bdia.2017006

[5]

Giuseppe Geymonat, Françoise Krasucki. Hodge decomposition for symmetric matrix fields and the elasticity complex in Lipschitz domains. Communications on Pure & Applied Analysis, 2009, 8 (1) : 295-309. doi: 10.3934/cpaa.2009.8.295

[6]

Huseyin Coskun. Nonlinear decomposition principle and fundamental matrix solutions for dynamic compartmental systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6553-6605. doi: 10.3934/dcdsb.2019155

[7]

Xianchao Xiu, Lingchen Kong. Rank-one and sparse matrix decomposition for dynamic MRI. Numerical Algebra, Control & Optimization, 2015, 5 (2) : 127-134. doi: 10.3934/naco.2015.5.127

[8]

Seung-Yeal Ha, Hansol Park. Emergent behaviors of the generalized Lohe matrix model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (8) : 4227-4261. doi: 10.3934/dcdsb.2020286

[9]

Markus Banagl. Singular spaces and generalized Poincaré complexes. Electronic Research Announcements, 2009, 16: 63-73. doi: 10.3934/era.2009.16.63

[10]

Sergei Avdonin, Fritz Gesztesy, Konstantin A. Makarov. Spectral estimation and inverse initial boundary value problems. Inverse Problems & Imaging, 2010, 4 (1) : 1-9. doi: 10.3934/ipi.2010.4.1

[11]

Gen Nakamura, Michiyuki Watanabe. An inverse boundary value problem for a nonlinear wave equation. Inverse Problems & Imaging, 2008, 2 (1) : 121-131. doi: 10.3934/ipi.2008.2.121

[12]

Sofian De Clercq, Wouter Rogiest, Bart Steyaert, Herwig Bruneel. Stochastic decomposition in discrete-time queues with generalized vacations and applications. Journal of Industrial & Management Optimization, 2012, 8 (4) : 925-938. doi: 10.3934/jimo.2012.8.925

[13]

Sonia Martínez, Jorge Cortés, Francesco Bullo. A catalog of inverse-kinematics planners for underactuated systems on matrix groups. Journal of Geometric Mechanics, 2009, 1 (4) : 445-460. doi: 10.3934/jgm.2009.1.445

[14]

Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, 2021, 15 (1) : 159-183. doi: 10.3934/ipi.2020076

[15]

Wei-guo Wang, Wei-chao Wang, Ren-cang Li. Deflating irreducible singular M-matrix algebraic Riccati equations. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 491-518. doi: 10.3934/naco.2013.3.491

[16]

Stefan Possanner, Claudia Negulescu. Diffusion limit of a generalized matrix Boltzmann equation for spin-polarized transport. Kinetic & Related Models, 2011, 4 (4) : 1159-1191. doi: 10.3934/krm.2011.4.1159

[17]

Guillaume Bal, Alexandre Jollivet. Generalized stability estimates in inverse transport theory. Inverse Problems & Imaging, 2018, 12 (1) : 59-90. doi: 10.3934/ipi.2018003

[18]

Hernan R. Henriquez. Generalized solutions for the abstract singular Cauchy problem. Communications on Pure & Applied Analysis, 2009, 8 (3) : 955-976. doi: 10.3934/cpaa.2009.8.955

[19]

Bastian Gebauer, Nuutti Hyvönen. Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem. Inverse Problems & Imaging, 2008, 2 (3) : 355-372. doi: 10.3934/ipi.2008.2.355

[20]

Panos K. Palamides, Alex P. Palamides. Singular boundary value problems, via Sperner's lemma. Conference Publications, 2007, 2007 (Special) : 814-823. doi: 10.3934/proc.2007.2007.814

 Impact Factor: 

Metrics

  • PDF downloads (34)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]