scholarly journals Some Rank Formulas for the Yang-Baxter Matrix Equation AXA = XAX

2021 ◽  
Vol 26 (6) ◽  
pp. 459-463
Author(s):  
Lifang DAI ◽  
Maolin LIANG ◽  
Yonghong SHEN
Keyword(s):  
Matrix Equation ◽  
Optimization Problem ◽  
Analytical Formulas ◽  
Square Matrix

Let A be an arbitrary square matrix, then equation AXA =XAX with unknown X is called Yang-Baxter matrix equation. It is difficult to find all the solutions of this equation for any given A . In this paper, the relations between the matrices A and B are established via solving the associated rank optimization problem, where B = AXA = XAX , and some analytical formulas are derived, which may be useful for finding all the solutions and analyzing the structures of the solutions of the above Yang-Baxter matrix equation.

Computing Solutions of the Yang-Baxter-like Matrix Equation for Diagonalisable Matrices

2015 ◽  
Vol 5 (1) ◽  
pp. 75-84 ◽  
Author(s):  
J. Ding ◽  
Noah H. Rhee
Keyword(s):  
Numerical Methods ◽  
Ergodic Theorem ◽  
Matrix Equation ◽  
Mean Ergodic Theorem ◽  
The Mean ◽  
Square Matrix

AbstractThe Yang-Baxter-like matrix equation AXA = XAX is reconsidered, where A is any complex square matrix. A collection of spectral solutions for the unknown square matrix X were previously found. When A is diagonalisable, by applying the mean ergodic theorem we propose numerical methods to calculate those solutions.

On the Solutions of the Matrix Equation f(X,X*)=g(X,X*)

1972 ◽  
Vol 15 (1) ◽  
pp. 45-49
Author(s):  
P. Basavappa
Keyword(s):  
Matrix Equation ◽  
Identity Matrix ◽  
Normal Matrices ◽  
Special Cases ◽  
The Matrix ◽  
Conjugate Transpose ◽  
Matrix Identities ◽  
Square Matrix

It is well known that the matrix identities XX*=I, X=X* and XX* = X*X, where X is a square matrix with complex elements, X* is the conjugate transpose of X and I is the identity matrix, characterize unitary, hermitian and normal matrices respectively. These identities are special cases of more general equations of the form (a)f(X, X*)=A and (b)f(Z, X*)=g(X, X*) where f(x, y) and g(x, y) are monomials of one of the following four forms: xyxy…xyxy, xyxy…xyx, yxyx…yxyx, and yxyx…yxy. In this paper all equations of the form (a) and (b) are solved completely. It may be noted a particular case of f(X, X*)=A, viz. XX'=A, where X is a real square matrix and X' is the transpose of X was solved by WeitzenbÖck [3]. The distinct equations given by (a) and (b) are enumerated and solved.

Further Solutions of a Yang-Baxter-like Matrix Equation

2013 ◽  
Vol 3 (4) ◽  
pp. 352-362 ◽  
Author(s):  
Jiu Ding ◽  
Chenhua Zhang ◽  
Noah H. Rhee
Keyword(s):  
Canonical Form ◽  
Infinite Number ◽  
Matrix Equation ◽  
Number Of Solutions ◽  
Jordan Canonical Form ◽  
The Matrix ◽  
Square Matrix

AbstractThe Yang-Baxter-like matrix equation AXA = XAX is reconsidered, and an infinite number of solutions that commute with any given complex square matrix A are found. Our results here are based on the fact that the matrix A can be replaced with its Jordan canonical form. We also discuss the explicit structure of the solutions obtained.

scholarly journals Some non-commuting solutions of the Yang-Baxter-like matrix equation

2020 ◽  
Vol 18 (1) ◽  
pp. 948-969
Author(s):  
Duan-Mei Zhou ◽  
Hong-Quang Vu
Keyword(s):  
Characteristic Polynomial ◽  
Matrix Equation ◽  
Numerical Examples ◽  
Square Matrix

Abstract Let A be a square matrix satisfying {A}^{4}=A . We solve the Yang-Baxter-like matrix equation AXA=XAX to find some solutions, based on analysis of the characteristic polynomial of A and its eigenvalues. We divide the problem into small cases so that we can find the solution easily. Finally, in order to illustrate the results, two numerical examples are presented.

Exergy and sensibility analysis of each individual effect in a kraft multiple effect evaporator

TAPPI Journal ◽  
2019 ◽  
Vol 18 (10) ◽  
pp. 607-618
Author(s):  
JÉSSICA MOREIRA ◽  
BRUNO LACERDA DE OLIVEIRA CAMPOS ◽  
ESLY FERREIRA DA COSTA JUNIOR ◽  
ANDRÉA OLIVEIRA SOUZA DA COSTA
Keyword(s):  
Optimization Problem ◽  
Real Data ◽  
Kraft Pulping ◽  
Steam Temperature ◽  
Input Flow ◽  
Transfer Coefficients ◽  
Heat Transfer Coefficients ◽  
Multiple Effect ◽  
Sensibility Analysis ◽  
Exergetic Analysis

The multiple effect evaporator (MEE) is an energy intensive step in the kraft pulping process. The exergetic analysis can be useful for locating irreversibilities in the process and pointing out which equipment is less efficient, and it could also be the object of optimization studies. In the present work, each evaporator of a real kraft system has been individually described using mass balance and thermodynamics principles (the first and the second laws). Real data from a kraft MEE were collected from a Brazilian plant and were used for the estimation of heat transfer coefficients in a nonlinear optimization problem, as well as for the validation of the model. An exergetic analysis was made for each effect individually, which resulted in effects 1A and 1B being the least efficient, and therefore having the greatest potential for improvement. A sensibility analysis was also performed, showing that steam temperature and liquor input flow rate are sensible parameters.

Imaging through Scattering Media with a Learning Based Prior

2020 ◽  
Vol 2020 (14) ◽  
pp. 306-1-306-6
Author(s):  
Florian Schiffers ◽  
Lionel Fiske ◽  
Pablo Ruiz ◽  
Aggelos K. Katsaggelos ◽  
Oliver Cossairt
Keyword(s):  
Optimization Problem ◽  
Autonomous Driving ◽  
Scattering Media ◽  
Photon Scattering ◽  
Point Spread ◽  
Spread Function ◽  
Radiative Transport Equation ◽  
Spatio Temporal ◽  
Transient Imaging

Imaging through scattering media finds applications in diverse fields from biomedicine to autonomous driving. However, interpreting the resulting images is difficult due to blur caused by the scattering of photons within the medium. Transient information, captured with fast temporal sensors, can be used to significantly improve the quality of images acquired in scattering conditions. Photon scattering, within a highly scattering media, is well modeled by the diffusion approximation of the Radiative Transport Equation (RTE). Its solution is easily derived which can be interpreted as a Spatio-Temporal Point Spread Function (STPSF). In this paper, we first discuss the properties of the ST-PSF and subsequently use this knowledge to simulate transient imaging through highly scattering media. We then propose a framework to invert the forward model, which assumes Poisson noise, to recover a noise-free, unblurred image by solving an optimization problem.

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