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In linear algebra, an eigenvector or characteristic vector of a square matrix is a vector that does not change its direction under the associated linear transformation. In other words—if v is a vector that is not zero, then it is an eigenvector of a square matrix ''A'' if ''A''v is a scalar multiple of v. This condition could be written as the equation: :: where ''λ'' is a scalar known as the eigenvalue or characteristic value associated with the eigenvector v. Geometrically, an eigenvector corresponding to a real, nonzero eigenvalue points in a direction that is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. There is a correspondence between ''n'' by ''n'' square matrices and linear transformations from an ''n''-dimensional vector space to itself. For this reason, it is equivalent to define eigenvalues and eigenvectors using either the language of matrices or the language of linear transformations. ==Overview== If a two-dimensional space is visualized as a rubber sheet, a linear map with two eigenvectors and associated eigenvalues λ1 and λ2 may be envisioned as stretching/compressing the sheet simultaneously along the two directions of the eigenvectors with the factors given by the eigenvalues. Thereby only the directions of the eigenvectors do not change. For example, the sheet could be stretched by a factor λ1 along the ''x''-axis and λ2 along the ''y''-axis, assuming the eigendirections being given by the directions of the coordinate axes. In two dimensions, there can be two such independent stretching directions, but they do not have to be at right angles to each other. A rotation in two dimensions is a linear map with no eigenvectors, and a shear, as in the photo, has only one eigenvector, with eigenvalue 1. Other vectors besides eigenvectors change their directions, unless the two eigenvalues are equal, in which case all vectors are eigenvectors with that eigenvalue, yielding a magnification—i.e., a linear map that alters neither shape nor direction, but only magnitude. A reflection may be viewed as stretching a line perpendicular to the axis of reflection by a factor of −1 while stretching the axis of reflection by a factor of 1. For 3D rotations, the axis of rotation is an eigenvector of eigenvalue 1. A three-coordinate vector may be seen as an arrow in three-dimensional space starting at the origin. In that case, an eigenvector is an arrow whose direction is either preserved or exactly reversed after multiplication by . The corresponding eigenvalue determines how the length of the arrow is changed by the operation, and whether its direction is reversed or not, determined by whether the eigenvalue is negative or positive. In abstract linear algebra, these concepts are naturally extended to more general situations, where the set of real scalar factors is replaced by any field of scalars (such as algebraic or complex numbers); the set of Cartesian vectors is replaced by any vector space (such as the continuous functions, the polynomials or the trigonometric series), and multiplication of a vector by a matrix is replaced by any linear operator that maps vectors to vectors (such as the derivative from calculus). In such cases, the "vector" in "eigenvector" may be replaced by a more specific term, such as "eigenfunction", "eigenmode", "eigenface", or "eigenstate". Thus, for example, the exponential function is an eigenfunction of the derivative operator, , with eigenvalue , since its derivative is . The set of all eigenvectors of a matrix (or linear operator), each paired with its corresponding eigenvalue, is called the eigensystem of that matrix.〔William H. Press, Saul A. Teukolsky, William T. Vetterling, Brian P. Flannery (2007), (''Numerical Recipes: The Art of Scientific Computing'', Chapter 11: ''Eigensystems.'', pages=563–597. Third edition, Cambridge University Press. ISBN 9780521880688 )〕 Any nonzero scalar multiple of an eigenvector is also an eigenvector corresponding to the same eigenvalue. An eigenspace or characteristic space of a matrix is the set of all eigenvectors of corresponding to the same eigenvalue, together with the zero vector.〔Wolfram Research, Inc. (2010) (''Eigenvector'' ). Accessed on 2010-01-29.〕 An eigenbasis for is any basis for the set of all vectors that consists of linearly independent eigenvectors of . Not every matrix has an eigenbasis, but every symmetric matrix does. The prefix eigen- is adopted from the German word ''eigen'' for "own-", "unique to", "peculiar to", or "belonging to" in the sense of "idiosyncratic" in relation to the originating matrix. Eigenvalues and eigenvectors have many applications in both pure and applied mathematics. They are used in matrix factorization, in quantum mechanics, and in many other areas. 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「Eigenvalues and eigenvectors」の詳細全文を読む スポンサード リンク
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