Search results
Results From The WOW.Com Content Network
As a special case, for every n × n real symmetric matrix, the eigenvalues are real and the eigenvectors can be chosen real and orthonormal. Thus a real symmetric matrix A can be decomposed as =, where Q is an orthogonal matrix whose columns are the real, orthonormal eigenvectors of A, and Λ is a diagonal matrix whose entries are the ...
Every real symmetric matrix is thus, up to choice of an orthonormal basis, a diagonal matrix. If and are real symmetric matrices that commute, then they can be simultaneously diagonalized by an orthogonal matrix: [2] there exists a basis of such that every element of the basis is an eigenvector for both and . Every real symmetric matrix is ...
In numerical linear algebra, the Jacobi eigenvalue algorithm is an iterative method for the calculation of the eigenvalues and eigenvectors of a real symmetric matrix (a process known as diagonalization).
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 ...
A 2×2 real and symmetric matrix representing a stretching and shearing of the plane. The eigenvectors of the matrix (red lines) are the two special directions such that every point on them will just slide on them. The example here, based on the Mona Lisa, provides a simple illustration. Each point on the painting can be represented as a vector ...
Also, recall that a Hermitian (or real symmetric) matrix has real eigenvalues. It can be shown [9] that, for a given matrix, the Rayleigh quotient reaches its minimum value (the smallest eigenvalue of M) when is (the corresponding eigenvector).
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 the case that A is identified with a Hermitian matrix, the matrix of A * is equal to its conjugate transpose. (If A is a real matrix, then this is equivalent to A T = A, that is, A is a symmetric matrix.) This condition implies that all eigenvalues of a Hermitian map are real: To see this, it is enough to apply it to the case when x = y is ...