2012-№1(34) Article 17

E.Kh. Naziev, A.Kh. Naziev, G.I. Keleynikova

Homogeneous linear differential systems with constant coefficients and the problem of eigenvalues (part 2). p.167-210

UDC 517.925

It is well known that to solve a homogeneous linear differential system with a constant matrix A one must usually find the Jordan canonical form of the matrix A and to obtain a matrix P such that J = P-1AP. To find a matrix J one should rely on the theory of elementary devisors of the characteristic matrix А – Е, which triggers off the so called full problem of eigenvalues. This problem is fraught with difficulty. In 1969 R. Bellman argued that there were no simple ways of calculating eigenvalues and eigenvectors of large matrices. Almost nothing has changed since then. The article suggests a new approach to solution of the problem by first computing eigenvectors and further finding eigenvalues, which is the opposite of the traditionally applied procedure. Part 1 was mainly theoretical and part 2 provides practical illustrations.

homogeneous linear differential equations with constant coefficients, eigenvalues, eigenvec-tors, homogeneous linear group, infinitesimal operator, one-parameter subgroup.

References

1. Gantmaher, F.R. Teorija matric [Theory of matrices]. Moscow, Nauka Publ., 1967, 575 p.

2. Gjunter, N.M. Integrirovanie uravnenij pervogo porjadka v chastnyh proizvodnyh [Integration of the equations of the first order in partial derivatives]. Moscow, GTTI Publ., 1934, 359 p.

3. Danilina, N.I. Chislennye metody [Calculus of approximations]. Moscow, Vysshaja shkola [Higher school] Publ., 1976, 368 p.

4. Efimov, N.V. Linejnaja algebra i mnogomernaja geometrija [Linear algebra and multidimensional geometry]. Moscow, Nauka Publ., 1970, 527 p.

5. Naziev, Je.H. Fazovye portrety odnorodnyh dinamicheskih sistem vtorogo porjadka [Phase portraits of homogeneous dynamic systems of the second order]. Kiev, 1991, 34 p. (Dep. of UkrSRISTI 30.10.91, no. 1429-91).

6. Jejzenhart, L.P. Nepreryvnye gruppy preobrazovanij [Persistent transformation groups]. Moscow, GIIL Publ., 1947, 359 p.