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 st