Courant minimax principle

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,

[math]\displaystyle{ \lambda_k=\min\limits_C\max\limits_{{\| x\| =1}, {Cx=0}}\langle Ax,x\rangle, }[/math]

where [math]\displaystyle{ C }[/math] is any [math]\displaystyle{ (k-1)\times n }[/math] 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 [math]\displaystyle{ q(x)=\langle Ax,x\rangle }[/math], A being a real symmetric matrix, the largest eigenvalue is given by [math]\displaystyle{ \lambda_1 = \max_{\|x\|=1} q(x) = q(x_1) }[/math], where [math]\displaystyle{ x_1 }[/math] is the corresponding eigenvector. Also (in the maximum theorem) subsequent eigenvalues [math]\displaystyle{ \lambda_k }[/math] and eigenvectors [math]\displaystyle{ x_k }[/math] are found by induction and orthogonal to each other; therefore, [math]\displaystyle{ \lambda_k =\max q(x_k) }[/math] with [math]\displaystyle{ \langle x_j, x_k \rangle = 0, \ j\lt k }[/math].

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

• Min-max theorem
• Max–min inequality
• Rayleigh quotient

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