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

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,

where is any 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 , A being a real symmetric matrix, the largest eigenvalue is given by , where is the corresponding eigenvector. Also (in the maximum theorem) subsequent eigenvalues and eigenvectors are found by induction and orthogonal to each other; therefore, with .

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.

The minimax principle also generalizes to eigenvalues of positive self-adjoint operators on Hilbert spaces, where it is commonly used to study the Sturm–Liouville problem.

See also

References

  • Courant, Richard; Hilbert, David (1989), Method of Mathematical Physics, Vol. I, Wiley-Interscience, ISBN 0-471-50447-5 (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
  • Horn, Roger; Johnson, Charles (1985), Matrix Analysis, Cambridge University Press, p. 179, ISBN 978-0-521-38632-6
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