TY - JOUR
T1 - A direct solution to the stochastic inverse eigenvalue problem for complex-valued eigenspectra
AU - McDonald, André M.
AU - van Wyk, Michaël A.
AU - Chen, Guanrong
PY - 2023/2/1
Y1 - 2023/2/1
N2 - We present a direct solution to the problem of constructing a stochastic matrix with prescribed eigenspectrum, widely referred to as the stochastic inverse eigenvalue problem. The solution uses Markov state disaggregation to construct a Markov chain with the associated stochastic transition matrix possessing the required eigenspectrum. Existing solutions that follow the same approach are limited to constructing matrices with real-valued eigenspectra. The novel solution directly constructs matrices with complex-valued eigenspectra by applying a new disaggregation technique in tandem with a technique from a previous solution. Due to this generalization, the novel solution is able to successfully model physical systems from a larger family. Furthermore, the novel solution constructs the matrix in a finite and predetermined number of iterations, and without numerical approximation. The solution is demonstrated by deriving an expression for a set of 4×4 stochastic matrices sharing the same prescribed complex-valued eigenspectrum and indexed by a real parameter.
AB - We present a direct solution to the problem of constructing a stochastic matrix with prescribed eigenspectrum, widely referred to as the stochastic inverse eigenvalue problem. The solution uses Markov state disaggregation to construct a Markov chain with the associated stochastic transition matrix possessing the required eigenspectrum. Existing solutions that follow the same approach are limited to constructing matrices with real-valued eigenspectra. The novel solution directly constructs matrices with complex-valued eigenspectra by applying a new disaggregation technique in tandem with a technique from a previous solution. Due to this generalization, the novel solution is able to successfully model physical systems from a larger family. Furthermore, the novel solution constructs the matrix in a finite and predetermined number of iterations, and without numerical approximation. The solution is demonstrated by deriving an expression for a set of 4×4 stochastic matrices sharing the same prescribed complex-valued eigenspectrum and indexed by a real parameter.
KW - Inverse eigenvalue problem
KW - Inverse stochastic problem
KW - Markov state disaggregation
KW - Stochastic matrix
UR - http://www.scopus.com/inward/record.url?scp=85141919244&partnerID=8YFLogxK
UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-85141919244&origin=recordpage
U2 - 10.1016/j.laa.2022.11.005
DO - 10.1016/j.laa.2022.11.005
M3 - 21_Publication in refereed journal
VL - 658
SP - 262
EP - 282
JO - Linear Algebra and Its Applications
JF - Linear Algebra and Its Applications
SN - 0024-3795
ER -