CMMSE algorithms for constructing doubly stochastic matrices with the relative gain array (combined matrix) A circle A(-T)

[EN] The Combined matrix of a nonsingular matrix A is defined by phi(A)=A T where degrees means the Hadamard (entrywise) product. If the matrix A describes the relation between inputs and outputs in a multivariable process control, phi(A) describes the relative gain array (RGA) of the process and it defines the Bristol method (IEEE Trans Autom Control 1:133-134, 1966) often used for Chemical processes (McAvoy in Interaction analysis: principles and applications. Instrument Society of America, Pittsburgh, 1983; Papadourakis et al. in Ind Eng Chem Res 26(6):1259-1262, 1987; Wang et al. in Chem Eng Technol, 10.1002/ceat.201500202, 2016; Kariwala et al. in Ind Eng Chem Res 45(5):1751-1757, 10.1021/ie050790r, 2006; Golender et al. in J Chem Inf Comput Sci 21(4):196-204, 10.1021/ci00032a004, 1981). The combined matrix has been studied in several works such as Bru et al. (J Appl Math, 10.1155/2014/182354, 2014), Fiedler and Markham (Linear Algebra Appl 435:1945-1955, 2011) and Johnson and Shapiro (SIAM J Algebraic Discrete Methods 7:627-644, 1986). Since phi(A)=(cij) has the property of Sigma kcik=Sigma kckj=1,i,j, when phi(A)>= 0, phi(A) is a doubly stochastic matrix. In certain chemical engineering applications a diagonal of the RGA in wchich the entries are near 1 is used to determine the pairing of inputs and outputs for further design analysis. Applications of these matrices can be found in Communication Theory, related with the satellite-switched time division multiple-access systems, and about a doubly stochastic automorphism of a graph. In this paper we present new algorithms to generate doubly stochastic matrices with the Combined matrix using Hessenberg matrices in Sect.3 and orthogonal/unitary matrices in Sect.4. In addition, we discuss what kind of doubly stochastic matrices are obtained with our algorithms and the possibility of generating a particular doubly stochastic matrix by the map phi.This work has been supported by Spanish Ministerio de Economia y Competitividad Grants MTM2014-58159-P, MTM2017-85669-P and MTM2017-90682-REDT.Fuster Capilla, RR.; Gasso Matoses, MT.; Gimenez Manglano, MI. (2019). CMMSE algorithms for constructing doubly stochastic matrices with the relative gain array (combined matrix) A circle A(-T). Journal of Mathematical Chemistry. 57(7):1700-1709. https://doi.org/10.1007/s10910-019-01032-1S17001709577E. Bristol, On a new measure of interaction for multivariable process control. IEEE Trans. Autom. Control 1, 133–134 (1966)R.A. Brualdi, Some applications of doubly stochastic matrices. Linear Algebra Appl. 107, 77–100 (1988)R. Bru, M.T. Gassó, I. Giménez, M. Santana, Nonnegative combined matrices. J. Appl. Math. (2014). https://doi.org/10.1155/2014/182354J. Cremona, Letter to the Editor. Am. Math. Monthly 4, 757 (2014)M. Fiedler, Relations between the diagonal entries of two mutually inverse positive definite matrices. Czechosl. Math. J. 14, 39–51 (1964)M. Fiedler, T.L. Markham, Combined matrices in special classes of matrices. Linear Algebra Appl. 435, 1945–1955 (2011)F.R. Gantmacher, The theory of matrices (American Mathematical Society, Chelsea, 1960)V.E. Golender, V.V. Drboglav, A.B. Rosenblit, Graph potentials method and its application for chemical information processing. J. Chem. Inf. Comput. Sci. 21(4), 196–204 (1981). https://doi.org/10.1021/ci00032a004R.A. Horn, C.R. Johnson, Topics in matrix analysis (Cambridge University Press, Cambridge, 1991)C. Johnson, H. Shapiro, Mathematical aspects of the relarive gain array. SIAM J. Algebraic Discrete Methods 7, 627–644 (1986)V. Kariwala, S. Skogestad, J.F. Forbes, Relative gain array for norm-bounded uncertain systems. Ind. Eng. Chem. Res. 45(5), 1751–1757 (2006). https://doi.org/10.1021/ie050790rH. Liebeck, A. Osborne, The generation of all rational orthogonal matrices. Am. Math. Monthly 98(2), 131–133 (1991)T.J. McAvoy, Interaction Analysis: Principles and Applications (Instrument Society of America, Pittsburgh, 1983)B. Mourad, Generalization of some results concerning eigenvalues of a certain class of matrices and some applications. Linear Multilinear Algebra 61, 1234–1243 (2013)A. Papadourakis et al., Relative gain array for units in plants with recycle. Ind. Eng. Chem. Res. 26(6), 1259–1262 (1987)H. Wang et al., Design and control of extractive distillation based on an effective relative gain array. Chem. Eng. Technol. (2016). https://doi.org/10.1002/ceat.201500202M. Hovd, S. Skogestad, Pairing criteria for decentralized control of unstable plants. Ind. Eng. Chem. Res. 33(9), 2134–2139 (1994). https://doi.org/10.1021/ie00033a016H. Wang, Y. Li, S. Weiyi, Y. Zhang, J. Guo, C. Li, Design and control of extractive distillation based on effective relative gain array. Chem. Eng. Technol. 39, 1–9 (2016). https://doi.org/10.1002/ceat.20150020

Combined matrices of sign regular matrices

This is the author’s version of a work that was accepted for publication in Linear Algebra and its Applications. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Linear Algebra and its Applications, VOL 498, (2016). DOI 10.1016/j.laa.2014.12.010.The combined matrix of a nonsingular matrix A is the Hadamard (entry wise) product C(A) = A ◦ (A−1)T . Since each row and column sum of C(A) is equal to one, the combined matrix is doubly stochastic when it is nonnegative. In this work, we study the nonnegativity of the combined matrix of sign regular matrices, based upon their signature. In particular, a few coordinates of the signature ε of A play a crucial role in determining whether or not C(A) is nonnegative. © 2014 Elsevier Inc. All rights reserved.Research supported by Spanish DGI grant number MTM2010-18674.Bru García, R.; Gasso Matoses, MT.; Gimenez Manglano, MI.; Santana-De Asis, MDJ. (2016). Combined matrices of sign regular matrices. Linear Algebra and its Applications. 498:88-98. https://doi.org/10.1016/j.laa.2014.12.010S889849

Diagonal entries of the combined matrix of a totally negative matrix

[EN] The combined matrix of a nonsingular matrix A is the Hadamard (entrywise) product . This paper deals with the characterization of the diagonal entries of a combined matrix C(A) of a given nonsingular real matrix A. A partial answer describing the diagonal entries of C(A) in the positive definite case was given by Fiedler in 1964. Recently in 2011, Fiedler and Markham characterized the sequence of diagonal entries of the combined matrix C(A) for any totally positive matrix A when the size is 3. For this case, we characterize totally negative matrices and we find necessary and sufficient conditions for the sequence of diagonal entries of C(A), in both cases, symmetric and nonsymmetric.This work was supported by Spanish DGI [grant number MTM-2014-58159-P]; Dominican Republic FONDOCYT [grant number 2015-1D2-166].Bru García, R.; Gasso Matoses, MT.; Gimenez Manglano, MI.; Santana-De Asis, MDJ. (2017). Diagonal entries of the combined matrix of a totally negative matrix. Linear and Multilinear Algebra. 65(10):1971-1984. https://doi.org/10.1080/03081087.2016.1261079S19711984651

Nonnegative combined matrices

The combined matrix of a nonsingular real matrix is the Hadamard (entrywise) product ∘ (−1) . It is well known that row (column) sums of combined matrices are constant and equal to one. Recently, some results on combined matrices of different classes of matrices have been done. In this work, we study some classes of matrices such that their combined matrices are nonnegative and obtain the relation with the sign pattern of . In this case the combined matrix is doubly stochastic.The authors would like to thank the referees for their suggestions that have improved the reading of this paper. This research is supported by Spanish DGI (Grant no. MTM2010-18674).Bru García, R.; Gasso Matoses, MT.; Gimenez Manglano, MI.; Santana, M. (2014). Nonnegative combined matrices. Journal of Applied Mathematics. 2014. https://doi.org/10.1155/2014/182354S2014Fiedler, M., & Markham, T. L. (2011). Combined matrices in special classes of matrices. Linear Algebra and its Applications, 435(8), 1945-1955. doi:10.1016/j.laa.2011.03.054Horn, R. A., & Johnson, C. R. (1991). Topics in Matrix Analysis. doi:10.1017/cbo9780511840371Brualdi, R. A. (1988). Some applications of doubly stochastic matrices. Linear Algebra and its Applications, 107, 77-100. doi:10.1016/0024-3795(88)90239-xMourad, B. (2013). Generalization of some results concerning eigenvalues of a certain class of matrices and some applications. Linear and Multilinear Algebra, 61(9), 1234-1243. doi:10.1080/03081087.2012.746330Bru, R., Corral, C., Giménez, I., & Mas, J. (2008). Classes of general H-matrices. Linear Algebra and its Applications, 429(10), 2358-2366. doi:10.1016/j.laa.2007.10.030Fiedler, M., & Hall, F. J. (2012). G-matrices. Linear Algebra and its Applications, 436(3), 731-741. doi:10.1016/j.laa.2011.08.001Ando, T. (1987). Totally positive matrices. Linear Algebra and its Applications, 90, 165-219. doi:10.1016/0024-3795(87)90313-2Peña, J. M. (2003). On nonsingular sign regular matrices. Linear Algebra and its Applications, 359(1-3), 91-100. doi:10.1016/s0024-3795(02)00437-

The Hadamard Product of a Nonsingular General H--Matrix and Its Inverse Transpose Is Diagonaly Dominant

[EN] We study the combined matrix of a nonsingular H-matrix. Iese matrices can belong to two diRerent H-matrices classes: the most common, invertible class, and one particular class named mixed class. DiRerent results regarding diagonal dominance of the inverse matrix and the combined matrix of a nonsingular H-matrix belonging to the referred classes are obtained. We conclude that the combined matrix of a nonsingular H-matrix is always diagonally dominant and then it is an H-matrix. In particular, the combined matrix in the invertible class remains in the same class.Ie authors thank the referee for suggesting changes that have improved the presentation of the paper. Iis research was supported by Spanish DGI Grant no. MTM2014-58159-P.Bru García, R.; Gasso Matoses, MT.; Gimenez Manglano, MI.; Scott, JA. (2015). The Hadamard Product of a Nonsingular General H--Matrix and Its Inverse Transpose Is Diagonaly Dominant. Journal of Applied Mathematics. 2015:1-6. https://doi.org/10.1155/2015/264680S162015Fiedler, M., & Markham, T. L. (1988). An inequality for the hadamard product of an M-matrix and an inverse M-matrix. Linear Algebra and its Applications, 101, 1-8. doi:10.1016/0024-3795(88)90139-5Johnson, C. R., & Shapiro, H. M. (1986). Mathematical Aspects of the Relative Gain Array $( A \circ A^{ - T} )$. SIAM Journal on Algebraic Discrete Methods, 7(4), 627-644. doi:10.1137/0607069Fiedler, M., & Markham, T. L. (2011). Combined matrices in special classes of matrices. Linear Algebra and its Applications, 435(8), 1945-1955. doi:10.1016/j.laa.2011.03.054Fiedler, M. (2010). Notes on Hilbert and Cauchy matrices. Linear Algebra and its Applications, 432(1), 351-356. doi:10.1016/j.laa.2009.08.014Bru, R., Gassó, M. T., Giménez, I., & Santana, M. (2014). Nonnegative Combined Matrices. Journal of Applied Mathematics, 2014, 1-5. doi:10.1155/2014/182354Bru, R., Gassó, M. T., Giménez, I., & Santana, M. (2016). Combined matrices of sign regular matrices. Linear Algebra and its Applications, 498, 88-98. doi:10.1016/j.laa.2014.12.010Bristol, E. (1966). On a new measure of interaction for multivariable process control. IEEE Transactions on Automatic Control, 11(1), 133-134. doi:10.1109/tac.1966.1098266Horn, R. A., & Johnson, C. R. (1991). Topics in Matrix Analysis. doi:10.1017/cbo9780511840371Lynn, M. S. (1964). On the Schur product of H-matrices and non-negative matrices, and related inequalities. Mathematical Proceedings of the Cambridge Philosophical Society, 60(3), 425-431. doi:10.1017/s0305004100037932Johnson, C. R. (1977). A Hadamard product involving N-matrices. Linear and Multilinear Algebra, 4(4), 261-264. doi:10.1080/03081087708817160Berman, A., & Plemmons, R. J. (1994). Nonnegative Matrices in the Mathematical Sciences. doi:10.1137/1.9781611971262Bru, R., Corral, C., Giménez, I., & Mas, J. (2008). Classes of general H-matrices. Linear Algebra and its Applications, 429(10), 2358-2366. doi:10.1016/j.laa.2007.10.030Bru, R., Corral, C., Giménez, I., & Mas, J. (2009). Schur complement of generalH-matrices. Numerical Linear Algebra with Applications, 16(11-12), 935-947. doi:10.1002/nla.668Varga, R. S. (1976). On recurring theorems on diagonal dominance. Linear Algebra and its Applications, 13(1-2), 1-9. doi:10.1016/0024-3795(76)90037-9Bru, R., Cvetković, L., Kostić, V., & Pedroche, F. (2010). Characterization of α1 and α2-matrices. Central European Journal of Mathematics, 8(1), 32-40. doi:10.2478/s11533-009-0068-6Cvetković, L. (2006). H-matrix theory vs. eigenvalue localization. Numerical Algorithms, 42(3-4), 229-245. doi:10.1007/s11075-006-9029-

Diagonal entries of the combined matrix of sign regular matrices of arder three

[EN] The study of the diagonal entries of the combined matrix of a nonsingular matrix A has been considered by different authors for the classes of Mmatrices, positive definite matrices and totally positive (negative) matrices. This problem appears to be difficult as the results have been done only for matrices of order three. In this work, we continue to give the characterization ofthe diagonal entries ofthe combined matrix ofthe remainder sign regular matrices. Thus, the problem is closed for all possible sign regular matrices of order three.Supported by Spanish Ministerio de Economía y Competitividad grants MTM2017-85669-P and MTM2017-90682-REDT and Dominican Republic FONDOCYT grant number 2015-1D2-166Gasso Matoses, MT.; Gil, I.; Gimenez Manglano, MI.; Santana-De Asis, MDJ.; Segura, E. (2021). Diagonal entries of the combined matrix of sign regular matrices of arder three. Proyecciones. 40(1):255-271. https://doi.org/10.22199/issn.0717-6279-2021-01-0016S25527140