Definition, Betydelse & Synonymer | Engelska ordet EIGENVALUE

Definition av EIGENVALUE

1. (matematik) egenvärde

Exempel på hur man kan använda EIGENVALUE i en mening

• However this last fact can be proved in an elementary way as follows: the eigenvalues of a real skew-symmetric matrix are purely imaginary (see below) and to every eigenvalue there corresponds the conjugate eigenvalue with the same multiplicity; therefore, as the determinant is the product of the eigenvalues, each one repeated according to its multiplicity, it follows at once that the determinant, if it is not 0, is a positive real number.
• Hermitian matrices also appear in techniques like singular value decomposition (SVD) and eigenvalue decomposition.
• In general, an eigenvector of a linear operator D defined on some vector space is a nonzero vector in the domain of D that, when D acts upon it, is simply scaled by some scalar value called an eigenvalue.
• Spectral methods can be used to solve differential equations (PDEs, ODEs, eigenvalue, etc) and optimization problems.
• In algebraic graph theory it equals the multiplicity of 0 as an eigenvalue of the Laplacian matrix of a finite graph.
• Indeed, if G acts on a vector space V over F, each simultaneous eigenspace for every element of G, if such exists, determines a multiplicative character on G: the eigenvalue on this common eigenspace of each element of the group.
• It is also used in eigenvalue algorithms (such as Rayleigh quotient iteration) to obtain an eigenvalue approximation from an eigenvector approximation.
• The primitive equations may be linearized to yield Laplace's tidal equations, an eigenvalue problem from which the analytical solution to the latitudinal structure of the flow may be determined.
• The diagonal entries of the normal form are the eigenvalues (of the operator), and the number of times each eigenvalue occurs is called the algebraic multiplicity of the eigenvalue.
• Other names include Wishart ensemble (in random matrix theory, probability distributions over matrices are usually called "ensembles"), or Wishart–Laguerre ensemble (since its eigenvalue distribution involve Laguerre polynomials), or LOE, LUE, LSE (in analogy with GOE, GUE, GSE).
• The rate of decay of the transient modes are given by (the logarithm of) their eigenvalues; the eigenvalue one corresponds to infinite half-life.
• Jacobi eigenvalue algorithm — select a small submatrix which can be diagonalized exactly, and repeat.
• Jacobi eigenvalue algorithm, a method for calculating the eigenvalues and eigenvectors of a real symmetric matrix.
• The signature counts how many time-like or space-like characters are in the spacetime, in the sense defined by special relativity: as used in particle physics, the metric has an eigenvalue on the time-like subspace, and its mirroring eigenvalue on the space-like subspace.
• As any rotation matrix has a single real eigenvalue, which is equal to +1, the corresponding eigenvector defines the rotation axis.
• The above-mentioned Euler vector is the eigenvector of a rotation matrix (a rotation matrix has a unique real eigenvalue).
• The adjugate is easier to compute explicitly than the inverse (though the inverse is easier to apply to a vector for problems that aren't small), and is more numerically sound because it remains well defined as the eigenvalue converges.
• For symmetric tridiagonal eigenvalue problems all eigenvalues (without eigenvectors) can be computed numerically in time O(n log(n)), using bisection on the characteristic polynomial.
• Reducing a general matrix to a Hessenberg matrix and then reducing further to a triangular matrix, instead of directly reducing a general matrix to a triangular matrix, often economizes the arithmetic involved in the QR algorithm for eigenvalue problems.
• Typically, the method is used in combination with some other method which finds approximate eigenvalues: the standard example is the bisection eigenvalue algorithm, another example is the Rayleigh quotient iteration, which is actually the same inverse iteration with the choice of the approximate eigenvalue as the Rayleigh quotient corresponding to the vector obtained on the previous step of the iteration.

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