On the inverse eigenproblem for symmetric and nonsymmetric arrowhead matrices. (English) Zbl 1455.65067
Summary: We present a new construction of a symmetric arrow matrix from a particular spectral information: let \(\lambda^{(n)}_1\) be the minimal eigenvalue of the matrix and \(\lambda_j^{(j)} \), \(j = 1, 2,\dots, n\) the maximal eigenvalues of all leading principal submatrices of the matrix. We use such a procedure to construct a nonsymmetric arrow matrix from the same spectral information plus to an eigenvector \(x^{(n)} = (x_1, x_2,\dots, x_n)\), so that \((x^{(n)}, \lambda_n^{(n)})\) is an eigenpair of the matrix. Moreover, our results generate an algorithmic procedure to compute a solution matrix.
MSC:
| 65F55 | Numerical methods for low-rank matrix approximation; matrix compression |
| 65F18 | Numerical solutions to inverse eigenvalue problems |
| 65F15 | Numerical computation of eigenvalues and eigenvectors of matrices |
| 15A18 | Eigenvalues, singular values, and eigenvectors |
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