| he inverse eigenvalue problem for matrices concerns the reconstruction of cor-responding matrices from some given eigenvalues or eigenvectors, its purpose is to construct specific structure matrices.The applications of inverse eigenvalue problems are extensive, it is widely used in many fields such as automatic control, system iden-tification, parametric identification, principal component analysis, structural design, remote sensing, survey, molecular spectrum analysis, quantum physics, solid mechan-ics, structural dynamics.In this paper, we consider four inverse eigenvalue problems as follows:Problem ?.Given real numbers ?1 <?2 < … <?n,?1<?2 <…<?n and a positive number ?,find a generalized Jacobi matrix Gn such that {?i}in=1 are the eigenvalues of Gn,{?i}i=1 are the eigenvalues of Jn1,where Jn-1 is the n-1 order leading principal submatrix of GnProblem ?.Given two different real numbers ?,?,and two linearly independent real vectors x = ?x1,x2,…?xn?T? Rn,y = ?y1,y2,…,yn?T ? Rn, find a generalized Jacobi matrix Gn such that Gnx = ?x, Gny =?y.Problem ?.Given real numbers ?1<?2 <…<?n-1,and real number ?or complex number??1,?2,…, ?n,and ?<0,find a pseudo-Jacobi matrix Tn such that {?i}in=1 are the eigenvalues of Tn,{?i}i=jn-1 are the eigenvalues of Jn1,where Jn-1 is the n-1 order leading principal submatrix of TnProblem ?.Given two different real numbers ?,?,and two linearly independent real vectors x=?x1,x2,…,xn?T ? Rn,y=?y1,y2,…,yn?T ?Rn,find a pseudo-Jacobi matrix Tn such that Tnx =?x, Tny=?y.For the above four problems, in this paper, we consider two classes of matrices about their features of eigenvalues and eigenvectors, necessary and sufficient condi-tions under which the problem is solvable are presented, and necessary and sufficient conditions under which the problem is uniquely solvable are also discussed, algorithm-s are provided to calculate corresponding solution , numerical examples are given to illustrate that corresponding algorithms are feasible and effective. |
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