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 (*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 *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 <*x*_{j},*x*_{k}> = 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.