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A Numerical Method for the Inverse Stochastic

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A Numerical Method for the Inverse Stochastic A NUMERICAL METHOD FOR THE INVERSE STOCHASTIC SPECTRUM PROBLEM y MOODY T CHU AND QUANLIN GUO Abstract Inverse sto chastic sp ectrum problem involves the construction of a sto chastic matrix with a prescrib ed sp ectrum A dierential equation aimed to bring forth the steep est descent ow in reducing the distance b etween isosp ectral matrices and nonnegative matrices represented in terms of some general co ordinates is describ ed The ow is further characterized by an analytic singular value decomp osition to maintain the numerical stability and to monitor the proximity to singularity This ow approachcan be used to design Markovchains with sp ecied structure Applications are demonstrated byn umerical examples Key words Sto chastic Matrix Least Squares Steep est Descent Isosp ectral Flow Structured Markovchain Analytic Singular Value Flow AMSMOS sub ject classications F H Intro duction Inverse eigenvalue problems concern the reconstruction of ma trices from prescrib ed sp ectral data The sp ectral data mayinvolve complete or partial information of eigenvalues or eigenvectors Generally a problem without any restric tions on the matrix is of little interest In order that the inverse eigenvalue problem b e meaningful it is often necessary to restrict the construction to sp ecial classes of matri ces such as symmetric To eplitz matrices or matrices with other sp ecial structures In this pap er we limit our attention to the so called sto chastic matrices ie matrices with nonnegative elements where all its row sums are equal to one We prop ose a numer ical pro cedure for the construction of a sto chastic matrix so that its sp ectrum agrees with a prescrib ed set of complex values If the set of prescrib ed values turns out to be infeasible the metho d pro duces a best approximation in the sense of least squares To our knowledge this inverse eigenvalue problem for sto chastic matrices has not b een studied extensively probably due to its diculty as we shall discuss below Neverthe less for a variety of physical problems that can be describ ed in the context of Markov chains an understanding of the inverse eigenvalue problem for sto chastic matrices and a capacity to solve the problem would make it p ossible to construct a system from its natural frequencies The metho d prop osed in the pap er app ears to be the rst attemp at tackling this problem numerically with some success Our technique can also be applied as a numerical way to solve the long standing inverse eigenvalue problems for nonnegative matrices Asso ciated with every inverse eigenvalue problem are two fundamental questions issue on solvability and the practical issue on computability The the theoretic ma jor eort in solvability has been to determine a necessary or a sucient condition under which an inverse eigenvalue problem has a solution whereas the main concern Department of Mathematics North Carolina State University Raleigh NC This researchwas supp orted in part by the National Science Foundation under grant DMS y Department of Mathematics North Carolina State University Raleigh NC 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1 −0.5 0 0.5 1 Fig by the KarpelevicTheorem in computability has been to develop an algorithm by which knowing a priori that the given sp ectral data are feasible a matrix can be constructed numerically Both questions are dicult and challenging Searching through the literature wehavefound only a handful of inverse eigenvalue problems that have b een completely understo o d or solved The fo cus of this pap er is on the computability for sto chastic matrices For sto chastic matrices the inverse eigenvalue problem is particularly dicult as result on existence by Karp elevic can be seen from the involvement in the best known Karp elevic completely characterized the set of points in the complex n plane that are eigenvalues of sto chastic n n matrices In particular the region is n symmetric ab out the real axis It is contained within the unit circle and its intersections ab with the unit circle are p oints z e where a and b run over all integers satisfying a b n The b oundary of consists of these intersection p oints and of n curvilinear arcs connecting them in circular order These arcs are characterized by sp ecic parametric equations whose formulas can be found in For example a complex number is an eigenvalue for a sto chastic matrix if and only if it b elongs to a region such as the one shown in Figure Complicated though it may seem the Karp elevic theorem characterizes only one complex value a time and do es alues of the not provide further insights into when two or more points in are eigenv n same sto chastic matrix Minc distinctively called the problem we are considering where the entire sp ectrum is given the inverse spectrum problem It is known that the inverse eigenvalue problem for nonnegative matrices is virtually equivalent to that for sto chastic matrices For example a complex nonzero number is an eigenvalue of a nonnegative matrix with a p ositive maximal eigenvalue r if and only if r is an eigenvalue of a sto chastic matrix Our problem is much more complicated b ecause it involves the entire sp ectrum Fortunately based on the following theorem we can pro ceed our computation once a nonnegative matrix is found Theorem If A is a nonnegative matrix with positive maximal eigenvalue r and a positive maximal eigenvector x then D r AD is a stochastic matrix where D diagfx x g n Wethus should turn our attention to the inverse eigenvalue or sp ectrum problems for nonnegative matrices a sub ject that has received considerable interest in the litera ture Some necessary and a few sucient conditions on whether a given set of complex numbers could be the sp ectrum of a nonnegative matrix can be found for example in and the references contained therein Yet numerical metho ds for constructing such a matrix even if the sp ectrum is feasible still need to b e develop ed Some discussion can b e found in Regardless of all the eorts the inverse eigenvalue problem for nonnegative matrices has not been completely resolved to this date In an earlier pap er the rst author has develop ed an algorithm that can con struct symmetric nonnegative matrices with prescrib ed sp ectra by means of dierential equations Symmetry was needed there b ecause the techniques by then were for ows in the group of orthogonal matrices only Up on realizing the existence of an analytic singular value decomp osition ASVD for a real analytic path of matrices we are able to advance the techniques in to general matrices in this pap er This pap er is organized as follows We reformulate the inverse sto chastic sp ectrum problem as that of nding the shortest distance between isosp ectral matrices and non negative matrices In x we intro duce a general co ordinate system to describ e these twotyp es of matrices and discuss how this setting naturally leads to a steep est descent ow This approach generalizes what has b een done b efore but requires the inversion of matrices that is potentially dangerous In x we argue that the steep est descent o w is in fact analytic and hence an analytic singular value decomp osition exists We therefore are able to describ e the ow by a more stable vector eld We illustrate the application of this dierential equation to the inverse sp ectrum problem by numerical examples in x Basic Formulation The given sp ectrum f g may b e complexvalued n It is not dicult to create a simple say tridiagonal realvalued matrix carrying the same sp ectrum For multiple eigenvalues one should also consider the p ossible real valued Jordan canonical form dep ending on the geometric multiplicity Matrices in the set nn M fP P jP R is nonsingularg obviously are isosp ectral to Let nn n fB B jB R g R denote the cone of all nonnegative matrices where means the Hadamard pro duct of n matrices Our basic idea is to nd the intersection of M and R Such an intersection if exists results in a nonnegative matrix isosp ectral to Furthermore if the condition in Theorem holds ie if the eigenvector corresp onding to the p ositive maximal eigenvalue is p ositive then we will have solved the inverse sp ectrum problem for sto chastic matrices by a diagonal similarity transformation The diculty as we have p ointed earlier is the lack of means to determine if the given sp ectrum is feasible An arbitrarily given set of values even if for all i may not be the n i n sp ectrum of any nonnegative matrix In this case it is reasonable to ask for only the b est p ossible approximation To handle b oth problems at the same time we reformulate the inverse sp ectrum problem as that of nding the shortest distance between M n and R minimize F P R kP P R R k where kk represents the Frob enius matrix norm Obviously if is feasible then F P R for some suitable P and R Note that the variable P in resides in the nn op en set of nonsingular matrices whereas R is simple a general matrix in R The optimization in sub jects to no other signicant constraint Since the optimization is over unb ounded op en domain it is p ossible that the minimum do es not exist We shall comment more on this point later The Frechet derivativeofF at P R acting on H K is calculated as follows F P RH K hP P R R H P P P HP K R R K i T T T T T T R R P Hi hP P R R P P P P P hP P R R R K i where h i denotes the Frob enius inner pro duct of two matrices Dene for abbrevia tion M P P P P R M P R R The norm of P R represents
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