An application of the Jordan canonical form to the epidemic problem

1980 ◽  
Vol 17 (2) ◽  
pp. 313-323 ◽  
Author(s):  
Linda P. Gilbert ◽  
A. M. Johnson
Keyword(s):  
Probability Distribution ◽  
Canonical Form ◽  
Difference Equations ◽  
Jordan Canonical Form ◽  
Stochastic Version ◽  
The Matrix

The stochastic version of the simple epidemic is considered in this paper. A method which has computational advantages is developed for computing the probability distribution of the process. The method is based on the Jordan canonical form of the matrix representing the system of differential-difference equations for the simple epidemic. It is also shown that the method can be applied to the general epidemic as well as any other right-shift process.

An application of the Jordan canonical form to the epidemic problem

1980 ◽  
Vol 17 (02) ◽  
pp. 313-323
Author(s):  
Linda P. Gilbert ◽  
A. M. Johnson
Keyword(s):  
Probability Distribution ◽  
Canonical Form ◽  
Difference Equations ◽  
Jordan Canonical Form ◽  
Stochastic Version ◽  
The Matrix

The stochastic version of the simple epidemic is considered in this paper. A method which has computational advantages is developed for computing the probability distribution of the process. The method is based on the Jordan canonical form of the matrix representing the system of differential-difference equations for the simple epidemic. It is also shown that the method can be applied to the general epidemic as well as any other right-shift process.

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 A New Solution to the Matrix EquationX−AX¯B=C

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Caiqin Song
Keyword(s):  
Unique Solution ◽  
Canonical Form ◽  
Matrix Equation ◽  
Explicit Solution ◽  
Symmetric Operator ◽  
Operator Matrix ◽  
Numerical Example ◽  
Controllability Matrix ◽  
The Matrix

We investigate the matrix equationX−AX¯B=C. For convenience, the matrix equationX−AX¯B=Cis named as Kalman-Yakubovich-conjugate matrix equation. The explicit solution is constructed when the above matrix equation has unique solution. And this solution is stated as a polynomial of coefficient matrices of the matrix equation. Moreover, the explicit solution is also expressed by the symmetric operator matrix, controllability matrix, and observability matrix. The proposed approach does not require the coefficient matrices to be in arbitrary canonical form. At the end of this paper, the numerical example is shown to illustrate the effectiveness of the proposed method.

scholarly journals A Jordan canonical form for reachable linear systems

1989 ◽  
Vol 122-124 ◽  
pp. 489-524 ◽  
Author(s):  
D. Hinrichsen ◽  
D. Prätzel-Wolters
Keyword(s):  
Linear Systems ◽  
Canonical Form ◽  
Jordan Canonical Form

scholarly journals Spectra inhabiting the left half-plane that are universally realizable

2021 ◽  
Vol 10 (1) ◽  
pp. 180-192
Author(s):  
Ricardo L. Soto
Keyword(s):  
Canonical Form ◽  
Half Plane ◽  
New Left ◽  
Nonnegative Matrix ◽  
Complex Numbers ◽  
Jordan Canonical Form ◽  
Positivity Condition ◽  
Left Half ◽  
Left Half Plane ◽  
New Criteria

Abstract Let Λ = {λ1, λ2, . . ., λ n } be a list of complex numbers. Λ is said to be realizable if it is the spectrum of an entrywise nonnegative matrix. Λ is universally realizable if it is realizable for each possible Jordan canonical form allowed by Λ. Minc ([21],1981) showed that if Λ is diagonalizably positively realizable, then Λ is universally realizable. The positivity condition is essential for the proof of Minc, and the question whether the result holds for nonnegative realizations has been open for almost forty years. Recently, two extensions of the Minc’s result have been proved in ([5], 2018) and ([12], 2020). In this work we characterize new left half-plane lists (λ1 > 0, Re λ i ≤ 0, i = 2, . . ., n) no positively realizable, which are universally realizable. We also show new criteria which allow to decide about the universal realizability of more general lists, extending in this way some previous results.

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