Hypermatrix of the Brain

  • Thomas L. Saaty
Conference paper


Decision-making, a natural and fundamental process of the brain, involves the use of pairwise comparisons. They are represented by a matrix whose entries belong to a fundamental scale, and from which an eigenvector of priorities that belongs to a ratio scale is derived. A simple decision is represented by a hierarchic structure, more complex ones by a feedback network. The alternatives from which the choice is made belong to the bottom level of the hierarchy whose upper levels contain the criteria and objectives of the decision. The derived eigenvectors are successively used to synthesize the outcome priorities by weighting and adding. A simple example of choosing the best school for the author’s son is used to illustrate this process. When there is dependence and feedback in a decision, synthesis requires the use of a stochastic supermatrix whose entries are block matrices of column normalized eigenvectors derived from paired comparisons. Stochasticity is ensured by also comparing the influence of the components that give rise to the blocks.

The synthesis of ratio scale signals in different parts of the brain leading to an overall state of awareness and feeling can be represented by a hypermatrix. The entries of this hypermatrix are block supermatrices each representing synthesis in one organ of the brain. The entries of a supermatrix are in turn blocks of matrices representing the contribution of a suborgan or part of an organ, whose columns are eigenfunctions of the general form [2]
$$w\left( z \right) = {z^{\ln \;b/\ln \;a}}P\left( {\ln \;z/\ln \;a} \right)$$
with a, b, and z complex. Here P(u) with u = ln z/ln a, is an arbitrary multivalued periodic function in u of period 1. Even without the multivaluedness of P, the function w(z) could be multivalued because In b/ln a is generally a complex number. If P is single-valued and In b/ln a turns out to be an integer or a rational number, then w(z) is a single-valued or finitely multivalued function, respectively. This generally multivalued solution is obtained in a way analogous to the real case. The solution represents the relative contribution of each member neuron of that suborgan to the synthesis of signals in the parent organ. The resulting hypermatrix is raised to powers (cycles signals) until stability is reached. Within each organ we have a different complex variable to represent synthesis within that organ. Feedback from control organs like the hypothalamus occurs in such a way as not to interfere with the variable itself but acts on muscles that reorient the activity of that organ. In the sense organs there must be a single valued analytic function, whereas in the feeling and perhaps also the perceiving organs we have a multivalued analytic function.


Paired Comparison Ratio Scale Bottom Level Stochastic Matrix Block Matrice 
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    Kuffler, S. J.G. Nichols: From Neuron to rain. Sunderland, MA, USA: Sinauer Associates (1976).Google Scholar
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    Saaty, T.L.: The Brain: Unraveling the Mystery of How it Works. 4922 Ellsworth Avenue, Pittsburgh, PA 15213, USA: RWS Publications (2000).Google Scholar
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    Saaty, T.L.: In the Brain Ratio Scales are Critical for Synthesis. Proceedings of the 5th ICANNGA. Prague, The Czech Republic: Academy of Sciences (2001).Google Scholar

Copyright information

© Springer-Verlag Wien 2001

Authors and Affiliations

  • Thomas L. Saaty
    • 1
  1. 1.University of PittsburghPittsburghUSA

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