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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,

λ k = min C max x = 1 , C x = 0 A x , x ,

where C is any (k − 1) × 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) = <Ax,x>, A being a real symmetric matrix, the largest eigenvalue is given by λ1 = max||x||=1q(x) = q(x1), where x1 is the corresponding eigenvector. Also (in the maximum theorem) subsequent eigenvalues λk and eigenvectors xk are found by induction and orthogonal to each other; therefore, λk = max q(xk) with <xj,xk> = 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.

References

Courant minimax principle Wikipedia


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