The Combined matrix of a nonsingular matrix A is defined by ϕ(A)=A∘A-1T\documentclass[12pt]{minimal}
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\begin{document}$$\phi (A)=A\circ \left( A^{-1}\right) ^T$$\end{document} where ∘\documentclass[12pt]{minimal}
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\begin{document}$$\circ $$\end{document} means the Hadamard (entrywise) product. If the matrix A describes the relation between inputs and outputs in a multivariable process control, ϕ(A)\documentclass[12pt]{minimal}
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\begin{document}$$\phi (A)$$\end{document} 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, https://doi.org/10.1002/ceat.201500202, 2016; Kariwala et al. in Ind Eng Chem Res 45(5):1751–1757, https://doi.org/10.1021/ie050790r, 2006; Golender et al. in J Chem Inf Comput Sci 21(4):196-204, https://doi.org/10.1021/ci00032a004, 1981). The combined matrix has been studied in several works such as Bru et al. (J Appl Math, https://doi.org/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 ϕ(A)=(cij)\documentclass[12pt]{minimal}
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\begin{document}$$\phi (A)=(c_{ij})$$\end{document} has the property of ∑kcik=∑kckj=1,∀i,j\documentclass[12pt]{minimal}
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\begin{document}$$\sum _k c_{ik} =\sum _k c_{kj}=1,\forall i, j$$\end{document}, when ϕ(A)≥0\documentclass[12pt]{minimal}
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\begin{document}$$\phi (A)\ge 0$$\end{document}, ϕ(A)\documentclass[12pt]{minimal}
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\begin{document}$$\phi (A)$$\end{document} 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 ϕ\documentclass[12pt]{minimal}
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\begin{document}$$\phi $$\end{document}.
