Commentary on "Expansions for Eigenfunctions and Eigenvalues of large-n Toeplitz Matrices"
نویسندگان
چکیده
منابع مشابه
Expansions for Eigenfunction and Eigenvalues of large-n Toeplitz Matrices
This note starts from work done by Dai, Geary, and Kadanoff[1] on exact eigenfunctions for Toeplitz operators. It builds methods for finding convergent expansions for eigenvectors and eigenvalues of singular, largen Toeplitz matrices, using the infinite-n case[1] as a starting point. One expansion is derived from operator equations having a two-dimensional continuous spectrum of right eigenvalu...
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The eigenvalues of a nonhermitian Toeplitz matrix A are usually highly sensitive to perturbations, having condition numbers that increase exponentially with the dimension N. An equivalent statement is that the resolvent ( ZZ A)’ of a Toeplitz matrix may be much larger in norm than the eigenvalues alone would suggest-exponentially large as a function of N, even when z is far from the spectrum. B...
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The matrix Tn[/] = (C„_y), 5, 7 = 0, 1, •• -, re is called the wth finite section of the infinite Toeplitz matrix (C,-¡) associated with the function f(9). We will be concerned with functions f(6) satisfying Condition A. Let f(d) be real, continuous and periodic with period 2w. Let min f(B) =/(0) = 0 and let 6 = 0 be the only value of 0 (mod 2tt) for which this minimum is attained. Condition A(...
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Let f be a nonnegative integrable function on [-r,r], T.Cf) the Ca+l) X(n+l) Toeplitz matrix associated with f and k,. its smallest eigenvalue. It is shown that the convergence of )t,. to rain f(O) can be exrmnentiallv fast even when f does not satisfy the smoothness condition of Kae, Murdoeh and Szeg6 (1953). Also a lower bound for ),,. corresponding to a large class of functions which do not ...
متن کاملEigenvalues of Hermitian Toeplitz matrices
The paper is concerned with finite Hermitian Toeplitz matrices whose entries in the first row grow like a polynomial. Such matrices cannot be viewed as truncations of an infinite Toeplitz matrix which is generated by an integrable function or a nice measure. The main results describe the first-order asymptotics of the extreme eigenvalues as the matrix dimension goes to infinity and also deliver...
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ژورنال
عنوان ژورنال: Papers in Physics
سال: 2010
ISSN: 1852-4249
DOI: 10.4279/pip.020004