Courant Minimax Principle - Introduction

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 λ₁ = max_||x||=1q(x) = q(x₁), where x₁ is the corresponding eigenvectors. 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,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.