The Inverse Eigenvalue Problems Of Two Structured Matrices

Structured inverse eigenvalue problems(SIEP) arise in a variety of applications. The list includes,but is not limited to,control design,solid mechanics, particle physics,mechanism design,system identification,mathematic.Generally speaking,the research of SIEP is concentrated on four fundamental questions:the theory of solvability,it has been to determine a necessary or a sufficient condition under which an inverse eigenvalue problem has a solution;the analysis of sensitivity,it concerns how the solution to an SIEP is modified by changes in the spectral data;the practice of computability,it is to develop a procedure by which,knowing a priori that the given spectral data are feasible,a matrix can be constructed in a numerically stable fashion;the reality of applicability,it is a matter of differentiation between whether the given data are exact or approximate, complete or incomplete,and whether an exact value or only an estimate of the parameters of the physical system is need.In this paper,two kinds of structure inverse eigenvalue problems are discussed. Firstly,an inverse eigenvalue problem of symmetric anti-diagonal matrices is discussed:a symmetric anti-diagonal matrix can be constructed if the following spectral data are known:an eigenvalue of C_n and the eigenvalues of C_n^[1] and the eigenvalues of C_n^[2],where C_n^[1] and C_n^[2] are n-1 submatrices of C_n, formed by deleting the([n/2]+1)th column and the([n/2]+1)th row or deleting the [n/2]th column and[n/2]th row of C_n respectively.As C_n has the same eigenvalues and nonzero elements as those of Jacobi matrix A_n,the inverse eigenvalue problem of C_n can be translated into the inverse eigenvalue problem of Jacobi matrix. Secondly,an inverse eigenvalue problem of anti-symmetric tri-diagonal matrix is discussed.That is,an anti-symmetric tri-diagonal matrix B can be constructed when a proper nonzero eigenvalue of matrix B and the module of eigenvalues of B_k,B_k-1,where B_k,B_k-1 are the n-1 submatrices of matrix A.B is constructed by the use of the separate relation of matrices and eigenpolynomial.Numerical experiments show that the algorithms are practical.