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Eigenform. In mathematics, an eigenform (meaning simultaneous Hecke eigenform with modular group SL (2, Z )) is a modular form which is an eigenvector for all Hecke operators Tm, m = 1, 2, 3, .... Eigenforms fall into the realm of number theory, but can be found in other areas of math and science such as analysis, combinatorics, and physics.
Eigen (C++ library) Eigen is a high-level C++ library of template headers for linear algebra, matrix and vector operations, geometrical transformations, numerical solvers and related algorithms. Eigen is open-source software licensed under the Mozilla Public License 2.0 since version 3.1.1. Earlier versions were licensed under the GNU Lesser ...
1) then v is an eigenvector of the linear transformation A and the scale factor λ is the eigenvalue corresponding to that eigenvector. Equation (1) is the eigenvalue equation for the matrix A . Equation (1) can be stated equivalently as (A − λ I) v = 0 , {\displaystyle \left(A-\lambda I\right)\mathbf {v} =\mathbf {0} ,} (2) where I is the n by n identity matrix and 0 is the zero vector ...
Eigendecomposition of a matrix. In linear algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way.
1) where λ is a scalar. The solutions to Equation (1) may also be subject to boundary conditions. Because of the boundary conditions, the possible values of λ are generally limited, for example to a discrete set λ 1 , λ 2 , … or to a continuous set over some range. The set of all possible eigenvalues of D is sometimes called its spectrum , which may be discrete, continuous, or a ...
The corresponding matrix of eigenvectors is unitary. The eigenvalues of a Hermitian matrix are real, since (λ − λ)v = (A* − A)v = (A − A)v = 0 for a non-zero eigenvector v. If A is real, there is an orthonormal basis for Rn consisting of eigenvectors of A if and only if A is symmetric.
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All eigenvalues of a unitary matrix. A {\displaystyle A} lie on the unit circle. Therefore, in this case, the spread is at most equal to the diameter of the circle, the number 2. The spread of a matrix depends only on the spectrum of the matrix (its multiset of eigenvalues). If a second matrix.