From: bumby@lagrange.rutgers.edu (Richard Bumby)
Newsgroups: sci.math
Subject: Re: Gershgorin circles
Date: 9 Oct 1998 19:18:02 -0400
Christophe Lermytte writes:
>Can someone state me the exact theorem on this ?
>Can you use this to estimate the roots of a polynomial ?
This is discussed in chapter 6 of Horn & Johnson, "Matrix Analysis",
taking almost 50 pages to describe, with references, variations and
extensions of the theorem.
To put the statement of the basic theorem in a memorable form, you
should look at the proof. The theorem is concerned with localizing
eigenvalues, so you note that every eigenvalue lambda of a matrix M
has an eigenvector v, and every vector v has an entry v_p of largest
absolute value. Rearrange Mv=lambda*v to relate v_p to the other
(smaller) entries of v. This shows that every eigenvalue differs from
some diagonal term by no more than the sum of the absolute values of
the other elements in that row.
If the matrix is diagonal, the diagonal elements are the eigenvalues,
and if you smoothly deform a general matrix to a diagonal matrix with
the same diagonal, the eigenvalues are also smoothly deformed (this
handwaving can be made both clear and rigorous, but takes a little space).
Some applications to bounds on roots of polynomials are given in the
reference. Other regions that isolate eigenvalues are also described.
Using only a companion matrix of a polynomial is not likely to give
practical bounds since all but one of the disks produced in the
theorem has center 0 and radius 1. However, you can use different
matrices to get information about the roots of the same polynomial
(either by conjugating to preserve eigenvalues while changing
eigenvectors or by taking polynomials to preserve eigenvectors while
applying a fixed polynomial to the eigenvalues).
--
R. T. Bumby ** Rutgers Math || Amer. Math. Monthly Problems Editor 1992--1996
bumby@math.rutgers.edu ||
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