Science and Science of Science (Ukraine), 1994, Poster, Berlin 1990.

A M A T R I X T H E O R Y O F I N F O R M A T I O N

In linear algebra matrices are operators transforming one vector into another: y = Mx. A matrix is a table with m rows and n columns which elements m have two indices, the first for rows, the second for columns. Rules for operations with matrices are well established, (addition, subtraction, multiplication, division and special matrix operations as transposition, diagonalization or inversion). Matrices already have many applications in information sciences.

Information is an operator acting on our thoughts and knowledge. If we imagine the knowledge as the vector x, then we can say that information is the matrix M. We can study its formal algebraic properties even if the vector x is inaccessible to us directly. E. G. we see the world and every picture is a matrix which elements are photons of different colors. Similarly, alphabet symbols can be written as the unit vectors e_j (0,0,...1,..., 0). Messages are then vector strings N in a multidimensional vector space [1]. We must know the ordering of symbols to get the meaning of a message.

It was deduced from the matrix formalism [2] that Boltzmann and Shannon H entropy functions are different and additive (Shannon H entropy function with binary logarithms is connected directly with indexing of symbols by a regular code [3]). They are logarithmic measures of symmetries of matrices. It is possible to permute independently rows and columns by multiplying matrices from the left and from the right by the unit permutation matrices P representing groups of cyclic permutations: P_mNP_n.

These two symmetries are separated in quadratic forms P^T_nN^TNP_n and P_mNN^TP^T_m. If an information matrix M is an incidence matrix, e.g. its rows are publications and columns their authors, these quadratic forms separate both kinds of elements.

Usually, mere symbols do not interest us, but their combinations, words, especially significant words, as key words, authors, references. They form subspaces in the original full space. Quadratic forms N^TN or simply column sums of original matrices are studied as statistics. Vector strings lead to points in vector space and we try to define positions of these points by analytical functions using right triangles between the position vector, the diagonal unit vector I multiplied by the arithmetical mean and the standard deviation vector. When in information matrices more nonzero elements appear (collective authorship, simultaneous citations) then singular values should be studied as hidden parameters [4].

Unnoticed the problem of inverse information elements remained. There are two kinds of inverse elements: additive a + b = 0; b = -a and multiplicative: axb = 1; b = 1/a. But at matrices, the problem of inverses is more complicated. Different matrices of coauthorships can be treated as matrices of graphs and therefore there exist inverse information matrices [5].

We can apply matrices on many levels and in many ways. Our world has fractal structure, systems of different sizes have similar topology. When we simplify, if W means the world matrix, then its inverse (interpreted as knowledge) W^-1 must satisfy, to be true, the formal equation W^-1W = I, where I is the identity matrix.

Problems connected with the matrix theory of information are technical, we need computers for evaluation of large information matrices and finding their eigenvalues and eigenvectors, and psychological. It is necessary to accept a new conception of the world, a notion, that we do not live and think in a 3 dimensional space but in a multidimensional one. We see only shadows on walls of a Plato's cave.

1. M. KUNZ, Information Processing in Linear Vector Space. Information Processing and Management, 20, (1984), 519.

2. M. KUNZ, Natural Vector Space. In: J. Mizerski, A. Posiewnik, J. Pykacz, M. Zukowski (Eds): Problems in Quantum Physics II, Gdansk', World Scientific, Singapore, (1990) p.377.

3. M. KUNZ, Entropies and Information Indices of Star Forests, Collection Czechoslovak Chemical Communications, 51(1986) 1856.

4. M. KUNZ, A Moebius Inversion of the Ulam Subgraphs Conjecture. Journal of Mathematical Chemistry 9, (1992), 297.

5. M. KUNZ, About Metrics of Bibliometrics, Journal Chemical Information and Computer Science, 33 (1993), 193.