# American Institute of Mathematical Sciences

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 and 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. [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. [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. [4] A. Torokhti and P. Howlett, "Computational Methods for Modelling of Nonlinear Systems," Elsevier, 2007. [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.

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##### References:
 [1] T. L. Boullion and P. L. Odell, "Generized Inverse Matrices," John Willey & Sons, Inc., New York, 1972. [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. [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. [4] A. Torokhti and P. Howlett, "Computational Methods for Modelling of Nonlinear Systems," Elsevier, 2007. [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.
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