Courant minimax principle
Encyclopedia
In mathematics, the Courant minimax principle gives the eigenvalues of a real symmetric matrix. It is named after Richard Courant
.
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 λk and eigenvectors xk are found by induction and orthogonal to each other; therefore, λk = max q(xk) with <x,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 space
s, where it is commonly used to study the Sturm–Liouville problem.
Richard Courant
Richard Courant was a German American mathematician.- Life :Courant was born in Lublinitz in the German Empire's Prussian Province of Silesia. During his youth, his parents had to move quite often, to Glatz, Breslau, and in 1905 to Berlin. He stayed in Breslau and entered the university there...
.
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 λk and eigenvectors xk are found by induction and orthogonal to each other; therefore, λk = max q(xk) with <x,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
Hypersphere
In mathematics, an n-sphere is a generalization of the surface of an ordinary sphere to arbitrary dimension. For any natural number n, an n-sphere of radius r is defined as the set of points in -dimensional Euclidean space which are at distance r from a central point, where the radius r may be any...
then the matrix A deforms that hypersphere into an ellipsoid. When the major axis on the intersecting hyperplane
Hyperplane
A hyperplane is a concept in geometry. It is a generalization of the plane into a different number of dimensions.A hyperplane of an n-dimensional space is a flat subset with dimension n − 1...
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 space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
s, where it is commonly used to study the Sturm–Liouville problem.