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

Abstract

[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.