Courant minimax principle explained

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,

λ _{k=min\limits} _{Cmax\limits} _{{\|} , {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) = \langle Ax,x\rangle , A being a real symmetric matrix, the largest eigenvalue is given by λ _{1} = max _{\|x\|=1} q(x) = q(x _{1)} , where x _{1} is the corresponding eigenvector. Also (in the maximum theorem) subsequent eigenvalues λ _{k} and eigenvectors x _{k} are found by induction and orthogonal to each other; therefore, λ _{k} =max q(x _{k)} with \langle x _{j,} x _{k} \rangle = 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.

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.