Algorithms for computing nth roots and the matrix sector function of nonsingular complex matrices

2000 ◽  
Author(s):  
M.A. Hasan ◽  
J.A.K. Hasan
Keyword(s):  
Complex Matrices ◽  
The Matrix ◽  
Sector Function

Eigenvalue localization for complex matrices

2014 ◽  
Vol 27 ◽  
Author(s):  
Ibrahim Gumus ◽  
Omar Hirzallah ◽  
Fuad Kittaneh
Keyword(s):  
Frobenius Norm ◽  
Complex Matrix ◽  
Complex Matrices ◽  
Eigenvalue Localization ◽  
The Matrix

Let $A$ be an $n\times n$ complex matrix with $n\geq 3$. It is shown that at least $n-2$ of the eigenvalues of $A$ lie in the disk \begin{equation*}\left\vert z-\frac{\func{tr}A}{n}\right\vert \leq \sqrt{\frac{n-1}{n}\left(\sqrt{\left( \left\Vert A\right\Vert _{2}^{2}-\frac{\left\vert \func{tr} A\right\vert ^{2}}{n}\right) ^{2}-\frac{\left\Vert A^{\ast }A-AA^{\ast}\right\Vert _{2}^{2}}{2}}-\frac{\limfunc{spd}\nolimits^{2}(A)}{2}\right) },\end{equation*} where $\left\Vert A\right\Vert _{2},$ $\func{tr}A$, and $\limfunc{spd}(A)$ denote the Frobenius norm, the trace, and the spread of $A$, respectively. In particular, if $A=\left[ a_{ij}\right] $ is normal, then at least $n-2$ of the eigenvalues of $A$ lie in the disk {\small \begin{eqnarray*} & & \left\vert z-\frac{\func{tr}A}{n}\right\vert \\ & & \leq \sqrt{\frac{n-1}{n}\left( \frac{\left\Vert A\right\Vert _{2}^{2}}{2}-\frac{\left\vert \func{tr}A\right\vert ^{2}}{n}-\frac{3}{2}\max_{i,j=1,\dots,n} \left( \sum_{\substack{ k=1 \\ k\neq i}}^{n}\left\vert a_{ki}\right\vert ^{2}+\sum_{\substack{ k=1 \\ k\neq j}}^{n}\left\vert a_{kj}\right\vert ^{2}+\frac{\left\vert a_{ii}-a_{jj}\right\vert ^{2}}{2}\right) \right) }. \end{eqnarray*}} Moreover, the constant $\frac{3}{2}$ can be replaced by $4$ if the matrix $A$ is Hermitian.

Determination of uranium and thorium in complex matrices by two solvent extraction separation techniques and photon electron rejecting alpha liquid spectrometry

2001 ◽  
Vol 89 (11-12) ◽  
Author(s):  
Marin Ayranov ◽  
K. Wacker ◽  
Urs Krähenbühl
Keyword(s):  
Solvent Extraction ◽  
Marine Sediments ◽  
Alpha Spectrometry ◽  
Complex Matrices ◽  
Separation Techniques ◽  
Extraction Separation ◽  
The Matrix ◽  
Matrix Components

The separation of uranium and thorium from complex matrixes such as marine sediments using solvent extraction and determination by means of Photon-Electron Rejecting Liquid Alpha Spectrometry (PERALS®) has successfully been performed. Two extraction schemes, using TOPO and HDEHP, respectively, were compared for the separation of uranium and thorium from the matrix components. The results show recoveries between 73 and 92% for

scholarly journals NEW SOLVABLE MATRIX INTEGRALS

2004 ◽  
Vol 19 (supp02) ◽  
pp. 276-293 ◽  
Author(s):  
A. YU. ORLOV
Keyword(s):  
Complex Matrices ◽  
Tau Function ◽  
Tau Functions ◽  
Matrix Argument ◽  
Matrix Integrals ◽  
The Matrix

We generalize the Harish-Chandra-Itzykson-Zuber and certain other integrals (the Gross-Witten integral, the integrals over complex matrices and the integrals over rectangle matrices) using a notion of the tau function of the matrix argument. In this case one can reduce multi-matrix integrals to integrals over eigenvalues, which in turn are certain tau functions. We also consider a generalization of the Kontsevich integral.

scholarly journals On the Yang-Baxter-like matrix equation for rank-two matrices

2017 ◽  
Vol 15 (1) ◽  
pp. 340-353 ◽  
Author(s):  
Duanmei Zhou ◽  
Guoliang Chen ◽  
Jiu Ding
Keyword(s):  
Matrix Equation ◽  
Full Column Rank ◽  
Complex Matrices ◽  
The Matrix ◽  
Quadratic Matrix Equation ◽  
Quadratic Matrix

Abstract Let A = PQT, where P and Q are two n × 2 complex matrices of full column rank such that QTP is singular. We solve the quadratic matrix equation AXA = XAX. Together with a previous paper devoted to the case that QTP is nonsingular, we have completely solved the matrix equation with any given matrix A of rank-two.

scholarly journals On the dense subsets of matrices with distinct eigenvalues and distinct singular values

2020 ◽  
Vol 36 (36) ◽  
pp. 834-846
Author(s):  
Himadri Lal Das ◽  
M. Rajesh Kannan
Keyword(s):  
Dense Subset ◽  
American Mathematical Society ◽  
Sufficient Conditions ◽  
Singular Values ◽  
Complex Matrices ◽  
Bounded Interval ◽  
The Matrix ◽  
Finite Set ◽  
Dense Sets ◽  
Necessary And Sufficient

It is well known that the set of all $n \times n$ matrices with distinct eigenvalues is a dense subset of the set of all real or complex $n \times n$ matrices. In [D.J. Hartfiel. Dense sets of diagonalizable matrices. {\em Proceedings of the American Mathematical Society}, 123(6):1669--1672, 1995.], the author established a necessary and sufficient condition for a subspace of the set of all $n \times n$ matrices to have a dense subset of matrices with distinct eigenvalues. The objective of this article is to identify necessary and sufficient conditions for a subset of the set of all $n \times n$ real or complex matrices to have a dense subset of matrices with distinct eigenvalues. Some results of Hartfiel are extended, and the existence of dense subsets of matrices with distinct singular values in the subsets of the set of all real or complex matrices is studied. Furthermore, for a matrix function $F(x)$, defined on a closed and bounded interval whose entries are analytic functions, it is proved that the set of all points for which the matrix $F(x)$ has repeated analytic eigenvalues/analytic singular values is either a finite set or the whole domain of the function $F$.  

scholarly journals Regions of convergence of a Padé family of iterations for the matrix sector function and the matrix pth root

2012 ◽  
Vol 236 (17) ◽  
pp. 4410-4420 ◽  
Author(s):  
Oleksandr Gomilko ◽  
Dmitry B. Karp ◽  
Minghua Lin ◽  
Krystyna Ziȩtak
Keyword(s):  
The Matrix ◽  
Sector Function
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