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If λ 1 (B) is the maximum eigenvalue of B and λ n (A) the minimum eigenvalue of A, a sufficient criterion to have A ≥ B is that λ n (A) ≥ λ 1 (B). If A or B is a multiple of the identity matrix, then this criterion is also necessary. The Loewner order does not have the least-upper-bound property, and therefore does not form a lattice.
In mathematics, the spectrum of a matrix is the set of its eigenvalues. [ 1 ] [ 2 ] [ 3 ] More generally, if T : V → V {\displaystyle T\colon V\to V} is a linear operator on any finite-dimensional vector space , its spectrum is the set of scalars λ {\displaystyle \lambda } such that T − λ I {\displaystyle T-\lambda I} is not invertible .
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, in particular linear algebra, the Bunch–Nielsen–Sorensen formula, [1] named after James R. Bunch, Christopher P. Nielsen and Danny C. Sorensen, expresses the eigenvectors of the sum of a symmetric matrix and the outer product, , of vector with itself.
The numerical range includes, in particular, the diagonal entries of the matrix (obtained by choosing x equal to the unit vectors along the coordinate axes) and the eigenvalues of the matrix (obtained by choosing x equal to the eigenvectors). In engineering, numerical ranges are used as a rough estimate of eigenvalues of A.
Let A be a square n × n matrix with n linearly independent eigenvectors q i (where i = 1, ..., n).Then A can be factored as = where Q is the square n × n matrix whose i th column is the eigenvector q i of A, and Λ is the diagonal matrix whose diagonal elements are the corresponding eigenvalues, Λ ii = λ i.
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