Publication

MATHICSE Technical Report : Bounds for the decay of the entries in inverses and Cauchy- Stieltjes functions of sparse, normal matrices

2017
Working Papers

Abstract

It is known that in many functions of banded, and more generally, sparse Hermitian positive definite matrices, the entries exhibit a rapid decay away from the sparsity pattern. This is in particular true for the inverse, and based on results for the inverse, bounds for Cauchy–Stieltjes functions of Hermitian positive definite matrices have recently been obtained. We add to the known results by considering the more general case of normal matrices, for which fewer and typically less satisfactory results exist so far. Starting from a very general estimate based on approximation properties of Chebyshev polynomials on ellipses, we obtain as special cases insightful decay bounds for various classes of normal matrices, including (shifted) skew- Hermitian and Hermitian indefinite matrices. In addition, some of our results improve over known bounds when applied to the Hermitian positive definite case.

Official source

https://infoscience.epfl.ch/entities/publication/af31adf2-60f0-4a11-83d7-eedc5f461142

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2017
Working Papers

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/af31adf2-60f0-4a11-83d7-eedc5f461142

About this result

Related concepts (32)

Hermitian matrix

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: or in matrix form: Hermitian matrices can be understood as the complex extension of real symmetric matrices.

Definite matrix

In mathematics, a symmetric matrix with real entries is positive-definite if the real number is positive for every nonzero real column vector where is the transpose of . More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number is positive for every nonzero complex column vector where denotes the conjugate transpose of Positive semi-definite matrices are defined similarly, except that the scalars and are required to be positive or zero (that is, nonnegative).

Normal matrix

In mathematics, a complex square matrix A is normal if it commutes with its conjugate transpose A^: The concept of normal matrices can be extended to normal operators on infinite dimensional normed spaces and to normal elements in C-algebras. As in the matrix case, normality means commutativity is preserved, to the extent possible, in the noncommutative setting. This makes normal operators, and normal elements of C*-algebras, more amenable to analysis.

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MATHICSE Technical Report : Bounds for the decay of the entries in inverses and Cauchy- Stieltjes functions of sparse, normal matrices

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