Courant minimax principle

From Wikipedia, the free encyclopedia

The factual accuracy of this article is disputed.
Please see the relevant discussion on the talk page. (April 2008)

In mathematics, the Courant minimax principle gives the eigenvalues of a real symmetric matrix. It is named after Richard Courant.

[edit] Introduction

The Courant minimax principle gives a condition for finding the eigenvalues for a real symmetric matrix. The Courant minimax principle is as follows:

For any real symmetric matrix A,

$\lambda_k=\min\limits_C\max\limits_{\binom{\| x\| =1}{Cx=0}}\langle Ax,x\rangle$ ,

where C is any (k - 1) x n matrix.

Notice that the vector x is an eigenvector to the corresponding eigenvalue λ.

The Courant minimax principle is a result of the maximum theorem, which says that for q(x) = <Ax,x>, A being a real symmetric matrix, the largest eigenvalue is given by λ₁ = max_||x||=1q(x)= q(x₁), where x₁ is the corresponding eigenvectors. Also (in the maximum theorem) subsequent eigenvalues λ_k & eigenvectors x_kare found by induction and orthogonal to each other; therefore, λ_k = max q(x_k) with <x,x_k> = 0, j < k.

The Courant minimax principle, as well as the maximum principle, can be visualized by imagining that if ||x|| = 1 is a hypersphere then the matrix A deforms that hypersphere into an ellipsoid. When the major axis on the intersecting hyperplane are maximized — i.e., the length of the quadratic form q(x) is maximized — this is the eigenvector and its length is the eigenvalue. All other eigenvectors will be perpendicular to this.

[edit] References

Courant, Richard & Hilbert, David (1989), Method of Mathematical Physics, Vol. I, Wiley-Interscience, ISBN 0471504475 (Pages 31-34; in most textbooks the "maximum-minimum method" is usually credited to Rayleigh and Ritz, who applied the calculus of variations in the theory of sound.)
Keener, James P. Principles of Applied Mathematics: Transformation and Approximation. Cambridge: Westview Press, 2000. ISBN 0-7382-0129-4