Courant minimax principle

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,

: $\backslash lambda\_k=\backslash min\backslash limits\_C\backslash max\backslash limits\_\{\{\backslash |\; x\backslash |\; =1\},\; \{Cx=0\}\}\backslash langle\; Ax,x\backslash rangle,$

where $C$ is any $(k-1)\backslash times\; n$ 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 $q(x)=\backslash langle\; Ax,x\backslash rangle$, A being a real symmetric matrix, the largest eigenvalue is given by $\backslash lambda\_1\; =\; \backslash max\_\{\backslash |x\backslash |=1\}\; q(x)\; =\; q(x\_1)$, where $x\_1$ is the corresponding eigenvector. Also (in the maximum theorem) subsequent eigenvalues $\backslash lambda\_k$ and eigenvectors $x\_k$ are found by induction and orthogonal to each other; therefore, $\backslash lambda\_k\; =\backslash max\; q(x\_k)$ 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.

• Min-max theorem
• Max–min inequality
• Rayleigh quotient

• {{citation|last=Courant|first=Richard|first2=David|last2=Hilbert|authorlink2=David Hilbert|title=Method of Mathematical Physics, Vol. I|publisher=Wiley-Interscience|year=1989|isbn=0-471-50447-5}} (Pages 31–34; in most textbooks the "maximum-minimum method" is usually credited to Rayleigh and Ritz, who applied the calculus of variations in the theory of sound.)
• Keener, James P. Principles of Applied Mathematics: Transformation and Approximation. Cambridge: Westview Press, 2000. {{ISBN|0-7382-0129-4}}
• {{citation|last=Horn|first=Roger|first2=Charles|last2=Johnson|title=Matrix Analysis|publisher=Cambridge University Press|year=1985|isbn=978-0-521-38632-6|page=179}}

Category:Mathematical principles