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- Eigenvalues_and_eigenvectors abstract "An eigenvector of a square matrix is a non-zero vector that, when the matrix is multiplied by , yields a constant multiple of , the multiplier being commonly denoted by . That is:(Because this equation uses post-multiplication by , it describes a right eigenvector.)The number is called the eigenvalue of corresponding to .If 2D space is visualized as a piece of cloth being stretched by the matrix, the eigenvectors would make up the line along the direction the cloth is stretched in and the line of cloth at the center of the stretching, whose direction isn't changed by the stretching either. The eigenvalues for the first line would give the scale to which the cloth is stretched, and for the second line the scale to which it's tightened. A reflection may be viewed as stretching a line to scale -1 while shrinking the axis of reflection to scale 1. For 3D rotations, the eigenvectors form the axis of rotation, and since the scale of the axis is unchanged by the rotation, their eigenvalues are all 1.In analytic geometry, for example, a three-element 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 matrix multiplication 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. Any multiple of an eigenvector is also an eigenvector, with the same eigenvalue. An eigenspace of a matrix is the set of all eigenvectors with the same eigenvalue, together with the zero vector. 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 terms characteristic vector, characteristic value, and characteristic space are also used for these concepts. The prefix eigen- is adopted from the German word eigen for "self-" or "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.".
- Eigenvalues_and_eigenvectors thumbnail Mona_Lisa_eigenvector_grid.png?width=300.
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- Eigenvalues_and_eigenvectors id "p/e035150".
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- Eigenvalues_and_eigenvectors title "Eigen value".
- Eigenvalues_and_eigenvectors title "Eigen vector".
- Eigenvalues_and_eigenvectors title "Eigenvalue".
- Eigenvalues_and_eigenvectors subject Category:Abstract_algebra.
- Eigenvalues_and_eigenvectors subject Category:Articles_including_recorded_pronunciations.
- Eigenvalues_and_eigenvectors subject Category:Linear_algebra.
- Eigenvalues_and_eigenvectors subject Category:Mathematical_physics.
- Eigenvalues_and_eigenvectors subject Category:Matrix_theory.
- Eigenvalues_and_eigenvectors subject Category:Singular_value_decomposition.
- Eigenvalues_and_eigenvectors type Agent.
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- Eigenvalues_and_eigenvectors comment "An eigenvector of a square matrix is a non-zero vector that, when the matrix is multiplied by , yields a constant multiple of , the multiplier being commonly denoted by .".
- Eigenvalues_and_eigenvectors label "Autovettore e autovalore".
- Eigenvalues_and_eigenvectors label "Eigenvalues and eigenvectors".
- Eigenvalues_and_eigenvectors label "Eigenwaarde (wiskunde)".
- Eigenvalues_and_eigenvectors label "Eigenwertproblem".
- Eigenvalues_and_eigenvectors label "Valeur propre, vecteur propre et espace propre".
- Eigenvalues_and_eigenvectors label "Valor próprio".
- Eigenvalues_and_eigenvectors label "Vector propio y valor propio".
- Eigenvalues_and_eigenvectors label "Wektory i wartości własne".
- Eigenvalues_and_eigenvectors label "Собственный вектор".
- Eigenvalues_and_eigenvectors label "القيم الذاتية والمتجهات الذاتية".
- Eigenvalues_and_eigenvectors label "固有値".
- Eigenvalues_and_eigenvectors label "特征向量".
- Eigenvalues_and_eigenvectors sameAs Vlastní_číslo.
- Eigenvalues_and_eigenvectors sameAs Eigenwertproblem.
- Eigenvalues_and_eigenvectors sameAs Ιδιοτιμές_και_ιδιοδιανύσματα.
- Eigenvalues_and_eigenvectors sameAs Vector_propio_y_valor_propio.
- Eigenvalues_and_eigenvectors sameAs Valeur_propre,_vecteur_propre_et_espace_propre.
- Eigenvalues_and_eigenvectors sameAs Nilai_dan_Vektor_Eigen.
- Eigenvalues_and_eigenvectors sameAs Autovettore_e_autovalore.
- Eigenvalues_and_eigenvectors sameAs 固有値.
- Eigenvalues_and_eigenvectors sameAs 고윳값.
- Eigenvalues_and_eigenvectors sameAs Eigenwaarde_(wiskunde).
- Eigenvalues_and_eigenvectors sameAs Wektory_i_wartości_własne.
- Eigenvalues_and_eigenvectors sameAs Valor_próprio.
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- Eigenvalues_and_eigenvectors sameAs Q190524.
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