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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク principal axis theorem ： ウィキペディア英語版 principal axis theorem In the mathematical fields of geometry and linear algebra, a principal axis is a certain line in a Euclidean space associated to an ellipsoid or hyperboloid, generalizing the major and minor axes of an ellipse. The ''principal axis theorem'' states that the principal axes are perpendicular, and gives a constructive procedure for finding them. Mathematically, the principal axis theorem is a generalization of the method of completing the square from elementary algebra. In linear algebra and functional analysis, the principal axis theorem is a geometrical counterpart of the spectral theorem. It has applications to the statistics of principal components analysis and the singular value decomposition. In physics, the theorem is fundamental to the study of angular momentum. ==Motivation== The equations in the Cartesian plane R2: :$\backslash frac+\backslash frac=1$ :$-\backslash frac=1$ define, respectively, an ellipse and a hyperbola. In each case, the ''x'' and ''y'' axes are the principal axes. This is easily seen, given that there are no ''cross-terms'' involving products ''xy'' in either expression. However, the situation is more complicated for equations like :$5x^2+8xy+5y^2=1.$ Here some method is required to determine whether this is an ellipse or a hyperbola. The basic observation is that if, by completing the square, the expression can be reduced to a sum of two squares then it defines an ellipse, whereas if it reduces to a difference of two squares then it is the equation of a hyperbola: :$u(x,y)^2+v(x,y)^2=1\backslash qquad\backslash text$ :$u(x,y)^2-v(x,y)^2=1\backslash qquad\backslash text.$ Thus, in our example expression, the problem is how to absorb the coefficient of the cross-term 8''xy'' into the functions ''u'' and ''v''. Formally, this problem is similar to the problem of matrix diagonalization, where one tries to find a suitable coordinate system in which the matrix of a linear transformation is diagonal. The first step is to find a matrix in which the technique of diagonalization can be applied. The trick is to write the equation in the following form: :$5x^2+8xy+5y^2=$ \begin x&y \end \begin 5&4\\4&5 \end \begin x\\y \end =\mathbf^TA\mathbf where the cross-term has been split into two equal parts. The matrix ''A'' in the above decomposition is a symmetric matrix. In particular, by the spectral theorem, it has real eigenvalues and is diagonalizable by an orthogonal matrix (''orthogonally diagonalizable''). To orthogonally diagonalize ''A'', one must first find its eigenvalues, and then find an orthonormal eigenbasis. Calculation reveals that the eigenvalues of ''A'' are :$\backslash lambda\_1\; =\; 1,\backslash quad\; \backslash lambda\_2\; =\; 9$ with corresponding eigenvectors :$\backslash mathbf\_1\; =\; \backslash begin1\backslash \backslash -1\backslash end,\backslash quad\; \backslash mathbf\_2=\backslash begin1\backslash \backslash 1\backslash end.$ Dividing these by their respective lengths yields an orthonormal eigenbasis: :$\backslash mathbf\_1\; =\; \backslash begin1/\backslash sqrt\backslash \backslash -1/\backslash sqrt\backslash end,\backslash quad\; \backslash mathbf\_2=\backslash begin1/\backslash sqrt\backslash \backslash 1/\backslash sqrt\backslash end.$ Now the matrix ''S'' = (u2 ) is an orthogonal matrix, since it has orthonormal columns, and ''A'' is diagonalized by: :$A\; =\; SDS^\; =\; SDS^T\; =$ \begin 1/\sqrt&1/\sqrt\\ -1/\sqrt&1/\sqrt \end \begin 1&0\\ 0&9 \end \begin 1/\sqrt&-1/\sqrt\\ 1/\sqrt&1/\sqrt \end. This applies to the present problem of "diagonalizing" the equation through the observation that :$5x^2+8xy+5y^2=\backslash mathbf^TA\backslash mathbf=\; (S^T\backslash mathbf)^TD(S^T\backslash mathbf)=1\backslash left(\backslash frac\backslash left(\backslash begin1/\backslash sqrt\backslash \backslash -1/\backslash sqrt\backslash end\backslash right),\backslash quad$ E_2 = \text\left(\begin1/\sqrt\\1/\sqrt\end\right). To summarize: * The equation is for an ellipse, since both eigenvalues are positive. (Otherwise, if one were positive and the other negative, it would be a hyperbola.) * The principal axes are the lines spanned by the eigenvectors. * The minimum and maximum distances to the origin can be read off the equation in diagonal form. Using this information, it is possible to attain a clear geometrical picture of the ellipse: to graph it, for instance. 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「principal axis theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.