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 commuting solutions to the Yang-Baxter-like matrix equation for identity matrix minus special rank-two matrices

Filomat ◽  
2018 ◽  
Vol 32 (13) ◽  
pp. 4591-4609
Author(s):  
Hui-Hui Yin ◽  
Xiang Wang ◽  
Xiao-Bin Tang ◽  
Lei Chen
Keyword(s):  
Matrix Equation ◽  
Full Column Rank ◽  
Identity Matrix ◽  
Complex Matrices ◽  
Special Rank ◽  
Quadratic Matrix Equation ◽  
Quadratic Matrix

Let A=I-PQT, where P and Q are two n x 2 complex matrices of full column rank such that det(QTP)=0. We find all the commuting solutions of the quadratic matrix equation AXA = XAX.

Newton-Shamanskii Method for a Quadratic Matrix Equation Arising in Quasi-Birth-Death Problems

2014 ◽  
Vol 4 (4) ◽  
pp. 386-395
Author(s):  
Pei-Chang Guo
Keyword(s):  
Markov Chains ◽  
Stationary Distribution ◽  
Discrete Time ◽  
Matrix Equation ◽  
Nonnegative Solution ◽  
Minimal Nonnegative Solution ◽  
The Minimal Nonnegative Solution ◽  
Quadratic Matrix Equation ◽  
Quadratic Matrix ◽  
Birth Death

AbstractIn order to determine the stationary distribution for discrete time quasi-birth-death Markov chains, it is necessary to find the minimal nonnegative solution of a quadratic matrix equation. The Newton-Shamanskii method is applied to solve this equation, and the sequence of matrices produced is monotonically increasing and converges to its minimal nonnegative solution. Numerical results illustrate the effectiveness of this procedure.

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