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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 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Courant minimax principle」の詳細全文を読む スポンサード リンク
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