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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 ...
Thus one can only calculate the numerical rank by making a decision which of the eigenvalues are close enough to zero. Pseudo-inverse The pseudo inverse of a matrix A {\displaystyle A} is the unique matrix X = A + {\displaystyle X=A^{+}} for which A X {\displaystyle AX} and X A {\displaystyle XA} are symmetric and for which A X A = A , X A X ...
The case of θ = 0, φ ≠ 0 is called a simple rotation, with two unit eigenvalues forming an axis plane, and a two-dimensional rotation orthogonal to the axis plane. Otherwise, there is no axis plane. The case of θ = φ is called an isoclinic rotation, having eigenvalues e ±iθ repeated twice, so every vector is rotated through an angle θ.
Comparing this equation to equation , it follows immediately that a left eigenvector of is the same as the transpose of a right eigenvector of , with the same eigenvalue. Furthermore, since the characteristic polynomial of A T {\displaystyle A^{\textsf {T}}} is the same as the characteristic polynomial of A {\displaystyle A} , the left and ...
Once the eigenvalues are computed, the eigenvectors could be calculated by solving the equation (), = using Gaussian elimination or any other method for solving matrix equations. However, in practical large-scale eigenvalue methods, the eigenvectors are usually computed in other ways, as a byproduct of the eigenvalue computation.
The surviving diagonal elements, ,, are known as eigenvalues and designated with in the defining equation, which reduces to =. The resulting equation is known as eigenvalue equation . [ 5 ] The eigenvectors and eigenvalues are derived from it via the characteristic polynomial .
In numerical linear algebra, the QR 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 developed in the late 1950s by John G. F. Francis and by Vera N. Kublanovskaya, working independently.
The NLEVP collection of nonlinear eigenvalue problems is a MATLAB package containing many nonlinear eigenvalue problems with various properties. [ 6 ] The FEAST eigenvalue solver is a software package for standard eigenvalue problems as well as nonlinear eigenvalue problems, designed from density-matrix representation in quantum mechanics ...