Adjoint Mappings and Inverses of Matrices

2006 ◽  
Vol 13 (03) ◽  
pp. 421-432 ◽  
Author(s):  
Predrag Stanimirović ◽  
Stojan Bogdanović ◽  
Miroslav Ćirić
Keyword(s):  
Drazin Inverse ◽  
Complex Matrix ◽  
Determinantal Formula ◽  
Partial Case ◽  
Determinantal Representation ◽  
The Matrix

In this paper, we investigate a general and determinantal representation, and conditions for the existence of a nonzero {2}-inverse X of a given complex matrix A. We introduce a determinantal formula for X, representing its elements in terms of minors of order s = rank (X), 1 ≤ s ≤ r = rank (A), taken from the matrix A and two adequately selected matrices. In accordance with these results, we find restrictions of the adjoint mapping such that the set A{2} is equal to the union of their images. Minors of {2}-inverses are also investigated. Restrictions to the set of {1, 2}-inverses produce the known results from [1–3, 10]. Also, in a partial case, we get known results from [11] relative to the Drazin inverse.

A Rapid Approach for Calculating the Damped Eigenvalues of a Gas Turbine on a Minicomputer: Theory

1984 ◽  
Vol 106 (2) ◽  
pp. 239-249 ◽  
Author(s):  
E. J. Gunter ◽  
R. R. Humphris ◽  
H. Springer
Keyword(s):  
Gas Turbine ◽  
Characteristic Polynomial ◽  
Normal Modes ◽  
Complex Matrix ◽  
Large Computer ◽  
Mainframe Computer ◽  
The Matrix ◽  
Rigid Body Modes ◽  
Transfer Method ◽  
Matrix Transfer Method

The calculation of the damped eigenvalues of a large multistation gas turbine by the complex matrix transfer procedure may encounter numerical difficulties, even on a large computer due to numerical round-off errors. In this paper, a procedure is presented in which the damped eigenvalues may be rapidly and accurately calculated on a minicomputer with accuracy which rivals that of a mainframe computer using the matrix transfer method. The method presented in this paper is based upon the use of constrained normal modes plus the rigid body modes in order to generate the characteristic polynomial of the system. The constrained undamped modes, using the matrix transfer process with scaling, may be very accurately calculated for a multistation turbine on a minicomputer. In this paper, a five station rotor is evaluated to demonstrate the procedure. A method is presented in which the characteristic polynomial may be automatically generated by Leverrier’s algorithm. The characteristic polynomial may be directly solved or the coefficients of the polynomial may be examined by the Routh criteria to determine stability. The method is accurate and easy to implement on a 16 bit minicomputer.

Simultaneous Determination of Eight New Generation Amide Insecticide Residues in Complex Matrix Agri-food by Multi-walled Carbon Nanotube Cleanup and UPLC-MS/MS

2021 ◽  
Author(s):  
Tao Lin ◽  
Xinglian Chen ◽  
Li Wang ◽  
Haixian Fang ◽  
Maoxuan Li ◽  
...  
Keyword(s):  
Mass Spectrometry ◽  
Simultaneous Determination ◽  
High Performance ◽  
Rapid Determination ◽  
Complex Matrix ◽  
Limit Of Quantification ◽  
Relative Standard ◽  
The Matrix ◽  
Multi Walled Carbon Nanotubes

Abstract The simultaneous determination method of 8 amide pesticides by multi-walled carbon nanotubes (MWCNs) cleanup, combined with QuEChERS method and ultra-high performance liquid chromatography-triple quadrupole tandem mass spectrometry has been developed and successfully applied in complex matrix such as green onions, celery, leeks, citrus, lychees, avocado. The matric effect of MWCNs is optimized and compared with QuEChERS materials. The results show that MWCNs can effectively reduce the matrix effect in sample extraction. The mass spectrometry is optimized, through their chemical structure skeletons, the ESI+ and ESI- mode are simultaneously scanned in the method. The coefficient (r) is greater than 0.9990, the limit of quantification ranges from 0.03 to 0.80 μg/kg, the recovery rate ranges from 71.2% to 120%, and the relative standard deviation (RSD) ranges from 3.8% to 9.4%. The method is fast, simple, sensitive, and has good purification effect. It is suitable for the rapid determination of amide pesticides in complex matrix agri-food.

scholarly journals Generalized Inverses Estimations by Means of Iterative Methods with Memory

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 2
Author(s):  
Santiago Artidiello ◽  
Alicia Cordero ◽  
Juan R. Torregrosa ◽  
María P. Vassileva
Keyword(s):  
Iterative Methods ◽  
Generalized Inverses ◽  
Drazin Inverse ◽  
Nonsingular Matrix ◽  
Type Method ◽  
The Matrix ◽  
Nonlinear Matrix ◽  
First Time ◽  
With Memory ◽  
Theoretical Results

A secant-type method is designed for approximating the inverse and some generalized inverses of a complex matrix A. For a nonsingular matrix, the proposed method gives us an approximation of the inverse and, when the matrix is singular, an approximation of the Moore–Penrose inverse and Drazin inverse are obtained. The convergence and the order of convergence is presented in each case. Some numerical tests allowed us to confirm the theoretical results and to compare the performance of our method with other known ones. With these results, the iterative methods with memory appear for the first time for estimating the solution of a nonlinear matrix equations.

On Solutions of Inverse Problem for Hermitian Generalized Hamiltonian Matrices

2013 ◽  
Vol 860-863 ◽  
pp. 2727-2731
Author(s):  
Kai Fu Liang ◽  
Ming Jun Li ◽  
Ze Lin Zhu
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Design Automation ◽  
Sufficient Conditions ◽  
Matrix Equations ◽  
Complex Matrix ◽  
Linear Quadratic ◽  
Real Representation ◽  
The Matrix ◽  
Necessary And Sufficient

Hamiltonian matrices have many applications to design automation and autocontrol, in particular in the linear-quadratic autocontrol problem. This paper studies the inverse problems of generalized Hamiltonian matrices for matrix equations. By real representation of complex matrix, we give the necessary and sufficient conditions for the existence of a Hermitian generalized Hamiltonian solutions to the matrix equations, and then derive the representation of the general solutions.

scholarly journals Involvement of Crawling and Attached Ciliates in the Aggregation of Particles in Wastewater Treatment Plants

2008 ◽  
Vol 1 ◽  
pp. ASWR.S752 ◽  
Author(s):  
Lucía Arregui ◽  
María Linares ◽  
Blanca Pέrez-Uz ◽  
Almudena Guinea ◽  
Susana Serrano
Keyword(s):  
Wastewater Treatment ◽  
Extracellular Polymeric Substances ◽  
Wastewater Treatment Plants ◽  
Biological Activities ◽  
Feeding Strategies ◽  
Complex Matrix ◽  
Active Secretion ◽  
Latex Beads ◽  
The Matrix

The biological community in activated sludge wastewater plants is organized within this ecosystem as bioaggregates or flocs, in which the biotic component is embedded in a complex matrix comprised of extracellular polymeric substances mainly of microbial origin. The aim of this work is to study the role of different floc-associated ciliates commonly reported in wastewater treatment plants-crawling Euplotes and sessile Vorticella- in the formation of aggregates. Flocs, in experiments with ciliates and latex beads, showed more compactation and cohesion among particles than those in the absence of ciliates. Ciliates have been shown to contribute to floc formation through different mechanisms such as the active secretion of polymeric substances (extrusomes), their biological activities (movement and feeding strategies), or the cysts formation capacity of some species. Staining with lectins coupled to fluorescein showed that carbohydrate of the matrix contained glucose, manose, N-acetyl-glucosamine and galactose. Protein fraction revealed over the latex beads surfaces could probably be of bacterial origin, but nucleic acids represented an important fraction of the extracellular polymeric substances of ciliate origin.

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.

scholarly journals Approximating the Matrix Sign Function Using a Novel Iterative Method

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
F. Soleymani ◽  
P. S. Stanimirović ◽  
S. Shateyi ◽  
F. Khaksar Haghani
Keyword(s):  
Iterative Method ◽  
Asymptotical Stability ◽  
Complex Matrix ◽  
Initial Matrix ◽  
Matrix Sign Function ◽  
Sign Function ◽  
The Matrix ◽  
Square Complex

This study presents a matrix iterative method for finding the sign of a square complex matrix. It is shown that the sequence of iterates converges to the sign and has asymptotical stability, provided that the initial matrix is appropriately chosen. Some illustrations are presented to support the theory.

scholarly journals Perturbation Bound for the Drazin Inverse of the Matrix-Value Function

2019 ◽  
Vol 2019 ◽  
pp. 1-14
Author(s):  
Yonghui Qin ◽  
Zhenshu Xie ◽  
Xiaoji Liu
Keyword(s):  
Perturbation Analysis ◽  
Upper Bound ◽  
Value Function ◽  
Drazin Inverse ◽  
Matrix Equations ◽  
Perturbation Bound ◽  
The Matrix

The perturbation analysis of the differential for the Drazin inverse of the matrix-value function A(t)∈Cn×n is investigated. An upper bound of the Drazin inverse and its differential is also considered. Applications to the perturbation bound for the solution of the matrix-value function coefficients some matrix equations are given.

Pseudo-Tournament Matrices and Their Eigenvalues

2014 ◽  
Vol 4 (3) ◽  
pp. 205-221
Author(s):  
Chuanlong Wang ◽  
Xuerong Yong
Keyword(s):  
Directed Graph ◽  
Round Robin ◽  
Column Vector ◽  
Complex Matrix ◽  
The Matrix ◽  
Tournament Matrices ◽  
Class Of Matrices

AbstractA tournament matrix and its corresponding directed graph both arise as a record of the outcomes of a round robin competition. An n × n complex matrix A is called h-pseudo-tournament if there exists a complex or real nonzero column vector h such that A + A* = hh* − I. This class of matrices is a generalisation of well-studied tournament-like matrices such as h-hypertournament matrices, generalised tournament matrices, tournament matrices, and elliptic matrices. We discuss the eigen-properties of an h-pseudo-tournament matrix, and obtain new results when the matrix specialises to one of these tournament-like matrices. Further, several results derived in previous articles prove to be corollaries of those reached here.

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