Connection between the spectral problem for linear matrix pencils and some problems of algebra
- 21 Downloads
We suggest a method by which the solution of systems of nonlinear algebraic equations in one and two variables can be reduced to the spectral problem for linear pencils of two matrices and for a system of two matrix pencils of two matrices, respectively. This method is substantially different from the traditional methods of solution; at the same time it is useful for the study of the solvability and the determination of the number of solutions of such systems. We propose a modification of the well-known elimination method for the solution of nonlinear algebraic systems of two equations in two unknowns which provides a new approach to studying and solving the problem.
KeywordsTraditional Method Algebraic Equation Linear Matrix Spectral Problem Algebraic System
Unable to display preview. Download preview PDF.
- 1.D. K. Faddeev and V. N. Faddeeva, Computational Methods of Linear Algebra, W. H. Freeman (1963).Google Scholar
- 3.F. R. Gantmakher, Matrix Theory [in German], Chelsea Publ.Google Scholar
- 4.V. N. Kublanovskaya, “The analysis of singular matrix pencils,” J. Sov. Math.,23, No. 1 (1983).Google Scholar
- 6.V. N. Fadeeva, Yu. A. Kuznetsov, G. N. Grekova, and T. A. Dolzhenkova, Numerical Methods of Linear Algebra, Bibliography 1828–1974 [in Russian], Novosibirsk (1976).Google Scholar
- 7.D. K. Faddeev, V. N. Kublanovskaya, and V. N. Fadeeva, “Linear algebraic systems with rectangular matrices,” in: Modern Numerical Methods [in Russian], Vol. 1, Kiev (1966); Moscow (1968).Google Scholar
- 8.V. N. Kublanovskaya, “The spectral problem for polynomial matrix pencils,” J. Sov. Math.,28, No. 3 (1985) (this issue).Google Scholar
- 9.G. I. Rasputin, “The numerical solution of nonlinear algebraic equations,” Vestn. Leningr. Univ., No. 1, 66–70 (1977).Google Scholar
- 10.V. N. Kublanovskaya, “The solution of the spectral problem for matrix pencils,” in: Numerical Methods of Linear Algebra [in Russian], Novosibirsk (1977), pp. 40–50.Google Scholar
- 11.V. N. Kublanovskaya and T. Ya. Kon'kova, “Solution of the eigenvalue problem for a regular pencil λA0-A1 with degenerate matrices,” J. Sov. Math.,23, No. 1 (1983).Google Scholar
- 12.T. Ya. Kon'kova, “Solution of the eigenvalue problem for a regular pencil D(λ)=λA0-A1 using deflated subspaces,” J. Sov. Math.,28, No. 3 (1985) (this issue).Google Scholar