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Given an n × n square matrix A of real or complex numbers, an eigenvalue λ and its associated generalized eigenvector v are a pair obeying the relation [1] =,where v is a nonzero n × 1 column vector, I is the n × n identity matrix, k is a positive integer, and both λ and v are allowed to be complex even when A is real.l When k = 1, the vector is called simply an eigenvector, and the pair ...
For defective matrices, the notion of eigenvectors generalizes to generalized eigenvectors and the diagonal matrix of eigenvalues generalizes to the Jordan normal form. Over an algebraically closed field, any matrix A has a Jordan normal form and therefore admits a basis of generalized eigenvectors and a decomposition into generalized eigenspaces .
The eigenvalues are real. The eigenvectors of A −1 are the same as the eigenvectors of A. Eigenvectors are only defined up to a multiplicative constant. That is, if Av = λv then cv is also an eigenvector for any scalar c ≠ 0. In particular, −v and e iθ v (for any θ) are also eigenvectors.
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 ...
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 λ {\displaystyle \lambda } forms the generalized eigenspace for λ {\displaystyle \lambda } .
Let = be an positive matrix: > for ,.Then the following statements hold. There is a positive real number r, called the Perron root or the Perron–Frobenius eigenvalue (also called the leading eigenvalue, principal eigenvalue or dominant eigenvalue), such that r is an eigenvalue of A and any other eigenvalue λ (possibly complex) in absolute value is strictly smaller than r, |λ| < r.
Matrix calculations can be often performed with different techniques. Many problems can be solved by both direct algorithms and iterative approaches. For example, the eigenvectors of a square matrix can be obtained by finding a sequence of vectors x n converging to an eigenvector when n tends to infinity. [43]
It is used in all applications that involve approximating eigenvalues and eigenvectors, often under different names. In quantum mechanics , where a system of particles is described using a Hamiltonian , the Ritz method uses trial wave functions to approximate the ground state eigenfunction with the lowest energy.