Direct Solution of the Stochastic Inverse Eigenvalue Problem for Complex-Valued Eigenspectra

Abstract

<p>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.</p>

The Construction of Ergodic Maps with Prescribed Invariant Density and Power Spectrum Poles

“…In this paper we use a recently proposed analytical solution to the StIEP [11] that directly constructs matrices with prescribed complex-valued eigenspectra. This solution generalizes an earlier solution [16] limited to constructing matrices with real-valued eigenspectra.…”

“…The StIEP solution of [11] uses the single-element matrix P = [1], which possesses the unity eigenvalue, as a starting point. This matrix is repeatedly enlarged over a predetermined number of rounds, thereby producing a stochastic matrix with the prescribed eigenspectrum.…”

“…We distinguish between two disaggregation techniques, referred to as the original and novel techniques (these were first proposed in [16] and [11], respectively). Whereas the original technique permits the insertion of a single real eigenvalue into the eigenspectrum of P, the novel technique permits the insertion of a complex conjugate pair of eigenvalues from the set C \ R (i.e., the strictly complex eigenvalues).…”

“…Whereas the original technique permits the insertion of a single real eigenvalue into the eigenspectrum of P, the novel technique permits the insertion of a complex conjugate pair of eigenvalues from the set C \ R (i.e., the strictly complex eigenvalues). These techniques are used in tandem by the solution of [11] to insert the prescribed eigenvalues into the eigenspectrum of the singleelement matrix over the number of rounds required, thereby producing a stochastic matrix that possesses the prescribed complex-valued eigenspectrum 1 .…”

“…We propose a method that utilizes the relationship outlined above to construct an ergodic and PWL hat map with prescribed statistics. The method constructs a stochastic matrix (i.e., a matrix with nonnegative entries and columns that sum to unity) with selectable eigenspectrum analytically using a recent solution to the inverse eigenvalue problem [11]. The hat map is then constructed such that its FPM equals this matrix.…”