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An example of an eigenvalue equation where the transformation is represented in terms of a differential operator is the time-independent Schrödinger equation in quantum mechanics: H ψ E = E ψ E {\displaystyle H\psi _{E}=E\psi _{E}\,}
In power iteration, for example, the eigenvector is actually computed before the eigenvalue (which is typically computed by the Rayleigh quotient of the eigenvector). [11] In the QR algorithm for a Hermitian matrix (or any normal matrix), the orthonormal eigenvectors are obtained as a product of the Q matrices from the steps in the algorithm ...
For example, a projection is a square matrix P satisfying P 2 = P. The roots of the corresponding scalar polynomial equation, λ 2 = λ, are 0 and 1. Thus any projection has 0 and 1 for its eigenvalues. The multiplicity of 0 as an eigenvalue is the nullity of P, while the multiplicity of 1 is the rank of P.
For example, if has real-valued elements, then it may be necessary for the eigenvalues and the components of the eigenvectors to have complex values. [35] [36] [37] The set spanned by all generalized eigenvectors for a given forms the generalized eigenspace for . [38]
Both of these differential equations are eigenvalue equations with eigenvalue E. As shown in an earlier example, the solution of Equation is the exponential = /. Equation is the time-independent Schrödinger equation.
In numerical linear algebra, the Arnoldi iteration is an eigenvalue algorithm and an important example of an iterative method.Arnoldi finds an approximation to the eigenvalues and eigenvectors of general (possibly non-Hermitian) matrices by constructing an orthonormal basis of the Krylov subspace, which makes it particularly useful when dealing with large sparse matrices.
In mathematics, an eigenvalue perturbation problem is that of finding the eigenvectors and eigenvalues of a system = that is perturbed from one with known eigenvectors and eigenvalues =. This is useful for studying how sensitive the original system's eigenvectors and eigenvalues x 0 i , λ 0 i , i = 1 , … n {\displaystyle x_{0i},\lambda _{0i ...
In numerical linear algebra, the Jacobi eigenvalue algorithm is an iterative method for the calculation of the eigenvalues and eigenvectors of a real symmetric matrix (a process known as diagonalization).