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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 .
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. [11]
In the special case of being a normal matrix, and thus also square, the spectral theorem ensures that it can be unitarily diagonalized using a basis of eigenvectors, and thus decomposed as = for some unitary matrix and diagonal matrix with complex elements along the diagonal.
The random walk normalized Laplacian can also be called the left normalized Laplacian := + since the normalization is performed by multiplying the Laplacian by the normalization matrix + on the left. It has each row summing to zero since P = D + A {\displaystyle P=D^{+}A} is right stochastic , assuming all the weights are non-negative.
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 ...
Notation: The index j represents the jth eigenvalue or eigenvector. The index i represents the ith component of an eigenvector. Both i and j go from 1 to n, where the matrix is size n x n. Eigenvectors are normalized. The eigenvalues are ordered in descending order.
For a normal matrix A (and only for a normal matrix), the eigenvectors can also be made orthonormal (=) and the eigendecomposition reads as =. In particular all unitary , Hermitian , or skew-Hermitian (in the real-valued case, all orthogonal , symmetric , or skew-symmetric , respectively) matrices are normal and therefore possess this property.
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, =.