Normal matrix

Summary

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. The spectral theorem states that a matrix is normal if and only if it is unitarily similar to a diagonal matrix, and therefore any matrix A satisfying the equation AA = AA is diagonalizable. The converse does not hold because diagonalizable matrices may have non-orthogonal eigenspaces. The left and right singular vectors in the singular value decomposition of a normal matrix \mathbf{A} = \mathbf{U} \boldsymbol{\Sigma} \mathbf{V}^* differ only in complex phase from each other an

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Related publications (3)

Bounds for the decay of the entries in inverses and Cauchy-Stieltjes functions of certain sparse, normal matrices

Marcel Schweitzer

2018

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

Marcel Schweitzer

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.

MATHICSE2017

MATHICSE Technical Report : A reduced basis approach to large-scale pseudospectra computation

Petar Sirkovic

For studying spectral properties of a non-normal matrix A ∈ Cn×n, information about its spectrum σ(A) alone is usually not enough. Effects of perturbations on σ(A) can be studied by computing ε-pseudospectra, that is the level-sets of the resolvent norm function g(z) = ‖(zI − A)−1‖2. The computation of ε-pseudospectra requires determining the smallest singular values σmin(zI − A) on for all z on a portion of the complex plane. In this work, we propose a reduced basis approach to pseudospectra computation that provides highly accurate estimates of pseudospectra in the region of interest. It incorporates the sampled singular vectors of zI − A for different values of z and implicitly exploits their smoothness properties. It provides rigorous upper and lower bounds for the pseudospectra in the region of interest. We also present a comparison of our approach to several existing approaches on a number of numerical examples, showing that our approach provides significant improvement in terms of computational time.

MATHICSE2016

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