Courant minimax principle

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,
 * $$\lambda_k=\min\limits_C\max\limits_{{\| x\| =1}, {Cx=0}}\langle Ax,x\rangle,$$

where $$C$$ is any $$(k-1)\times n$$ 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 $$\lambda_1 = \max_{\|x\|=1} q(x) = q(x_1)$$, 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, $$\lambda_k =\max q(x_k)$$ 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 &mdash; i.e., the length of the quadratic form q(x) is maximized &mdash; 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.