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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]
For a matrix, eigenvalues and eigenvectors can be used to decompose the matrix—for example by diagonalizing it. Eigenvalues and eigenvectors give rise to many closely related mathematical concepts, and the prefix eigen-is applied liberally when naming them:
Intuitively, we glob together the Jordan block invariant subspaces corresponding to the same eigenvalue. In the extreme case where A is a multiple of the identity matrix we have k = n and l = 1. The projection onto Y i and along all the other Y j ( j ≠ i) is called the spectral projection of A at v i and is usually denoted by P(λ i ; A).
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 ...
As an example, the identity matrix commutes with all matrices, which between them do not all commute. If the set of matrices considered is restricted to Hermitian matrices without multiple eigenvalues, then commutativity is transitive, as a consequence of the characterization in terms of eigenvectors.
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 linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is Q T Q = Q Q T = I , {\displaystyle Q^{\mathrm {T} }Q=QQ^{\mathrm {T} }=I,} where Q T is the transpose of Q and I is the identity matrix .
The Gram matrix of any orthonormal basis is the identity matrix. Equivalently, the Gram matrix of the rows or the columns of a real rotation matrix is the identity matrix. Likewise, the Gram matrix of the rows or columns of a unitary matrix is the identity matrix.