Схема алгоритму покрокового приведення двох матриць у формі Шура до простого виду

Of great importance for modern technology are the issues of the dynamic behavior of machines and structures. Given the trends of today such as new information technologies, the transformation of design processes; creation of CAD systems (automation systems for design work), issues of improving methods for solving dynamics problems are of particular importance.Numerical modeling has become an integral part of the study of the most complex processes for the most complex physical models, and the Finite Element Method has become the main numerical method for their study. These trends will inevitably be accompanied by a significant increase in the number of parameters for determining the state of the object. In particular, in the dynamics of machines, a significant number of degrees of freedom. As a result, the appearance of the problem of a multiple increase in the size of the task. The solution of the eigenvalue problem (EigenValue) can be considered as an important component in the construction of numerical-analytical approaches that are alternative to simple step-by-step integration schemes of the Runge-Kutta type in large-scale problems. Certain benefits can be obtained if the linear transformation matrices are first reduced to a simple form. This approach is widely used in dynamics (modal analysis). In this paper, we propose a scheme for a step-by-step numerical analysis of the structure of the matrix K in a problem and a scheme for constructing a Jordan basis for the general case of roots of a characteristic polynomial (for real and complex roots, simple and multiple). As the starting form, the standard eigenvalue problem with the matrix K previously reduced to the Schur form (block-triangular shape matrix) is adopted. The scheme is accompanied by the solution of test cases.