Solvable and unsolvable problems. Using Procrustes analysis algorithm for obtaining a family of Hadamard matrices
Introduction: The development of the Hadamard matrix theory encountered an obstacle caused not so much by the nature of the integer problem as by the artificial limitation of the solution of quadratic equations applying exhaustive search algorithms. Ignoring the direct path and rejecting irrationality led to the opinion that the hypothesis of the existence of Hadamard matrices was unprovable. Purpose: To prove the solvability of the Hadamard problem by orthogonal matrices via identifying their stable connection with matrices containing irrational elements. Results: We show that irrationality manifests itself in the quadratic norm of the columns of the Hadamard matrix of the second order. We consider the transfer of iterative algorithms for calculating roots to the matrix. To minimize the maximum absolute value element of the orthogonal matrix we propose the Procrustes analysis algorithm. Since Hadamard matrices are determined by invariants of smaller-order matrices embedded in their structure, the algorithm turns out to be a universal basis for finding them together. We consider the hypothesis of the existence of Hadamard matrices in the operational domain of iterative algorithms determined over the field of real numbers that give advantage over the tools in the form of finite fields and groups. Practical relevance: Orthogonal sequences obtained from rows (columns) of Hadamard matrices, and high-order Hadamard matrices themselves are of great practical importance for problems of noise-correcting coding, compression, masking and image processing.
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Ministry of Education and Science of the Russian Federation