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, where C is any 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, A being a real symmetric matrix, the largest eigenvalue is given by , 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, with. 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.