Английская Википедия:Courant minimax principle

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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>\lambda_k=\min\limits_C\max\limits_Шаблон:\\langle Ax,x\rangle,</math>

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

References

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