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'',
: $\lambda_k=\min\limits_C\max\limits_\langle Ax,x\rangle,$

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

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

* (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.
*{{citation, last=Horn, first=Roger, first2=Charles, last2=Johnson, title=Matrix Analysis, publisher=Cambridge University Press, year=1985, isbn=978-0-521-38632-6, page=179

Mathematical principles