which an eigenvector is stretched or shrunk. If the eigenvalue is negative, the eigenvector's direction is reversed. The eigenvectors and eigenvalues of a...

canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way....

designing efficient and stable algorithms for finding the eigenvalues of a matrix. These eigenvalue algorithms may also find eigenvectors. Given an n × n...

momentum of 1 kg⋅m/s if and only if one of the eigenvalues of the momentum operator is 1 kg⋅m/s. The corresponding eigenvector (which physicists call an...

{\displaystyle x\in [0,L]} , the following are the eigenvalues and normalized eigenvectors. The eigenvalues are ordered in descending order. λ j = − j 2 π...

necessary for the eigenvalues and the components of the eigenvectors to have complex values. The set spanned by all generalized eigenvectors for a given λ...

}}_{k}+1\right)~.} Its eigenvalues are therefore 1 or −1. It may thus be utilized as an interaction term in a Hamiltonian, splitting the energy eigenvalues of its symmetric...

graph into eigenvalues and eigenvectors. Analogously to the classical Fourier transform, the eigenvalues represent frequencies and eigenvectors form what...

except positive multiples of v (respectively, left eigenvectors except ww'w), i.e., all other eigenvectors must have at least one negative or non-real component...

solving for the eigenvalues, the eigenvectors corresponding to the eigenvalues can be found by the defining equation. Two n × n matrices A and B are similar...

general eigenvectors is known. Moreover, the eigenvectors are not unique because any linear combination of eigenvectors for the same eigenvalue is also...

perturbation of a simple eigenvalue (see in multiplicity of eigenvalues). In the entry applications of eigenvalues and eigenvectors we find numerous scientific...

produces a response y(t) = λf(t), where λ is a complex scalar eigenvalue. Eigenvalues and eigenvectors Hilbert–Schmidt theorem Spectral theory of ordinary differential...

every vector is expressible as a linear combination of eigenvectors. In both cases, all eigenvalues are real. A symmetric n×n-matrix is called positive-definite...

matrix (or of the endomorphism), and there are, at most, n eigenvalues. If a basis exists that consists only of eigenvectors, the matrix of f on this basis...

algorithm or QR iteration is an eigenvalue algorithm: that is, a procedure to calculate the eigenvalues and eigenvectors of a matrix. The QR algorithm was...

is an eigenvalue algorithm and an important example of an iterative method. Arnoldi finds an approximation to the eigenvalues and eigenvectors of general...

the matrix of left eigenvectors. Rearrange the eigenvectors and eigenvalues Sort the columns of the eigenvector matrix V and eigenvalue matrix D in order...

linear algebra, the Jacobi eigenvalue algorithm is an iterative method for the calculation of the eigenvalues and eigenvectors of a real symmetric matrix...

fixed points can be analyzed by determining their respective eigenvalues and eigenvectors. Beginning with the Jacobian: ( 0 − 1 − 1 1 a 0 z 0 x − c ) {\displaystyle...

formulas for eigenvalues and eigenvectors of the second derivative can be obtained. For example, assuming x ∈ [ 0 , L ] {\displaystyle x\in [0,L]} and homogeneous...

for example, a square matrix is invertible if and only if it has a nonzero determinant and the eigenvalues of a square matrix are the roots of a polynomial...

{\displaystyle m} "most useful" (tending towards extreme highest/lowest) eigenvalues and eigenvectors of an n × n {\displaystyle n\times n} Hermitian matrix, where...

which we can use an eigenvalue algorithm. It is used in all applications that involve approximating eigenvalues and eigenvectors, often under different...

consisting of eigenvectors of T {\displaystyle T} , and the diagonal entries of D {\displaystyle D}  are the corresponding eigenvalues of T {\displaystyle...

modal matrix is used in the diagonalization process involving eigenvalues and eigenvectors. Specifically the modal matrix M {\displaystyle M} for the matrix...

effective dielectric tensor from which one calculates systems eigenvalues and eigenvectors. As usual, from this tensor, magneto-optical phenomena are described...

all eigenvectors corresponding to a particular eigenvalue of f forms a vector space known as the eigenspace corresponding to the eigenvalue (and f) in...

adjacency matrix is symmetric. The relationship between a graph and the eigenvalues and eigenvectors of its adjacency matrix is studied in spectral graph theory...

are real, and that A has n linearly independent eigenvectors. Moreover, a Hermitian matrix has orthogonal eigenvectors for distinct eigenvalues. Even if...