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Direct Solution of the Stochastic Inverse Eigenvalue Problem for Complex-Valued Eigenspectra
  • Andre McDonald ,
  • Anton van Wyk ,
  • Guanrong Chen
Andre McDonald
Defence and Security Cluster

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Anton van Wyk
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Guanrong Chen
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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 stochastic transition matrix possessing the required eigenspectrum. Existing solutions that follow the same approach are limited to constructing matrices with real-valued eigenspectra only. 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 x 4 stochastic matrices sharing the same prescribed complex-valued eigenspectrum and indexed by a real parameter.
Feb 2023Published in Linear Algebra and its Applications volume 658 on pages 262-282. 10.1016/j.laa.2022.11.005