Abstract
We consider the following inverse extreme eigenvalue problem: given the real numbers {λ1j,λjj}j=1n and the real vector x(n)=x1,x2,…,xn, to construct a nonsymmetric tridiagonal matrix and a nonsymmetric arrow matrix such that {λ1j,λjj}j=1n are the minimal and the maximal eigenvalues of each one of their leading principal submatrices, and x(n),λn(n) is an eigenpair of the matrix. We give sufficient conditions for the existence of such matrices. Moreover our results generate an algorithmic procedure to compute a unique solution matrix.
Highlights
We consider a particular inverse eigenvalue problem for real nonsymmetric tridiagonal matrices of the form a1 b1 c1 a2 b2 A = ( ( c2 a3 d ) ) ; d d bn−1 (1)bici > 0, i = 1, 2, . . . , n − 1, and for real nonsymmetric arrow matrices of the form a1 b1 b2 ⋅ ⋅ ⋅ bn−1 c1 a2 B =
To show the existence of a nonsymmetric tridiagonal matrix A with the required properties is equivalent to show that the system of equations
To show the existence of a nonsymmetric arrow matrix A with the required properties is equivalent to show that the system of equations
Summary
We consider a particular inverse eigenvalue problem for real nonsymmetric tridiagonal matrices of the form a1 b1 N − 1, and for real nonsymmetric arrow matrices of the form a1 b1 b2 ⋅ ⋅ ⋅ bn−1
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