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In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the i-th row and j-th column is equal to the complex conjugate of the element in the j-th row and i-th column, for all indices i and j: = ¯
For a Hermitian matrix, the norm squared of the jth component of a normalized eigenvector can be calculated using only the matrix eigenvalues and the eigenvalues of the corresponding minor matrix, |, | = (()) (), where is the submatrix formed by removing the jth row and column from the original matrix.
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 ...
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an complex matrix is an matrix obtained by transposing and applying complex conjugation to each entry (the complex conjugate of + being , for real numbers and ).
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 more general matrices, the QR algorithm yields the Schur decomposition first, from which the eigenvectors can be obtained by a backsubstitution procedure. [13])
In mathematics, the Rayleigh quotient [1] (/ ˈ r eɪ. l i /) for a given complex Hermitian matrix and nonzero vector is defined as: [2] [3] (,) =. For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the conjugate transpose to the usual transpose ′.
The eigenvectors may be found by the usual methods of linear algebra, but a convenient trick is to note that a Pauli spin matrix is an involutory matrix, that is, the square of the above matrix is the identity matrix. Thus a (matrix) solution to the eigenvector problem with eigenvalues of ±1 is simply 1 ± S u. That is,
In mathematics, an eigenvalue perturbation problem is that of finding the eigenvectors and eigenvalues of a system = that is perturbed from one with known eigenvectors and eigenvalues =. This is useful for studying how sensitive the original system's eigenvectors and eigenvalues x 0 i , λ 0 i , i = 1 , … n {\displaystyle x_{0i},\lambda _{0i ...