The lower boundfor the difference of the extreme eigenvalues of an n × n Hermitian block 2 × 2 matrix A =is established, and conditions necessary and sufficient for this bound to be attained at A are provided. Some corollaries of this result are derived. In part...
Let an n × n Hermitian matrix A be presented in block 2 × 2 form as \\(A = \\left[ {\\begin{array}{*{20}c} {A_{11} } & {A_{12} } \\\ {A_{12}^* } & {A_{22} } \\\ \\end{array} } ight]\\) , where A12≠ 0, and assume that the diagonal blocks A11...
First, by applying the results about the eigenvalue perturbation bounds for Hermitian block tridiagonal matrices in paper [1], we obtain a new efficient method to estimate the perturbation bounds for singular values of block tridiagonal matrix. Second, we consider the perturbation bounds for ...
(1993): Bounds for eigenvalues and singular values of matrix completions. Linear and Multilinear Algebra 33, 233-250I. Gohberg, L. Rodman, T. Shalom and H. Woerdeman, Bounds for eigenvalues and singular values of matrix completions, Linear and Multilinear Analysis 33 (1992), 233–249. Math...
The matrixShas the real eigenvalue as the first entry on the diagonal and the repeated eigenvalue represented by the lower right 2-by-2 block. The eigenvalues of the 2-by-2 block are also eigenvalues ofA: eig(S(2:3,2:3)) ans = ...
Theorem 1 (Eigenvector-eigenvalue identity) Let be an Hermitian matrix, with eigenvalues . Let be a unit eigenvector corresponding to the eigenvalue , and let be the component of . Then where is the Hermitian matrix formed by deleting the row and column from . When we posted the first ve...
An upper quasitriangular matrix can result from the Schur decomposition or generalized Schur (QZ) decomposition of a real matrix. An upper quasitriangular matrix is block upper triangular, with 1-by-1 and 2-by-2 blocks of nonzero values along the diagonal. ...
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The asymptotic distribution of singular values and eigenvalues of non-Hermitian block Toeplitz matrices is studied. These matrices are associated with the Fourier series of an univariate function f . The asymptotic distribution of singular values is computed when f belongs to L 2 and is matrix-value...
The eigenvalues of a matrix [M] are the values of λ such that: [M] v = λ v where: v = eigenvector λ = lambda = eigenvalue this gives: |M - λ I| = 0 where I = identity matrix this gives: = 0 so (m00- λ) (m11- λ) (m22- λ) + m01 m12 m20 + m02 m10 m21 ...