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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.
The three eigenvectors are ordered ,, by their eigenvalues ; [50] then is the primary orientation/dip of clast, is the secondary and is the tertiary, in terms of strength. The clast orientation is defined as the direction of the eigenvector, on a compass rose of 360° .
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 ...
When the eigenvalues (and eigenvectors) of a symmetric matrix are known, the following values are easily calculated. Singular values The singular values of a (square) matrix A {\displaystyle A} are the square roots of the (non-negative) eigenvalues of A T A {\displaystyle A^{T}A} .
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.
In mathematics, power iteration (also known as the power method) is an eigenvalue algorithm: given a diagonalizable matrix, the algorithm will produce a number , which is the greatest (in absolute value) eigenvalue of , and a nonzero vector , which is a corresponding eigenvector of , that is, =.
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 ...
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.