Construct Nonnegative Matrices With Specific Spectrum

The nonnegative inverse eigenvalue problem(NIEP)is the problem of finding neces-sary and sufficient conditions for a list of complex numbers to be the spectrum of some nonnegative matrix.NIEP has been studied in more than half a century and has made many achievements,but the NIEP with order greater than or equal to 5 is not solved com-pletely.In view of the complexity of studying NIEP,the following two aspects are often used to study the NIEP:studying NIEP often need to construct nonnegative matrices with specific structure,the construction of these nonnegative matrices with specific structure is more practical and good for theoretical research;In the process of studying NIEP,the complex set ? is often divided into some set of fewer elements,so we can use the method of combination spectrum to study the NIEP.Construction of nonnegative matrices with specific structures:symmetric matrices,peysymmetric matrices,stochastic matrices,double stochastic matrices,and so on.The properties of the special matrix are more convenient for the study of the NIEP.And in the partitioned spectrum,the set containing only real numbers and the set containing complex numbers are often discussed separately.In this paper,an algorithm for solving the NIEP with specific spectrum is proposed based on the theorem of combination spectrum and the good properties of the stochastic matrix.For solving a complex set ? with specific structure,2 or 3-order nonnegative generalized double stochastic matrices are generated,and then the 2 or 3-order nonnegative generalized double stochastic matrices are used as units in the construction process.The iterative algorithm combines the low-order matrices to reconstruct the spectral set and finally obtains the nonnegative generalized stochastic matrices with the spectrum ?.The advantages of the fast solving algorithm are:The generation of the 2 or 3-order nonnegative generalized double stochastic matrices is independent and then the parallelization can be considered in the design of the program;For the 2 or 3-order nonnegative generalized double stochastic matrix,the left and the right eigenvectors of the Perron eigenvalues can be obtained directly without the need for complex mathematical operations,to further reduce the error occurring in the calculation process;Estimated time complexity of the algorithm is a polynomial time,means that for high complex set also can be solved.