The real and the symmetric nonnegative inverse eigenvalue problems are different. (English) Zbl 0861.15007
Summary: We show that there exist real numbers \(\lambda_1, \lambda_2, \dots, \lambda_n\) that occur as the eigenvalues of an entry-wise nonnegative \(n\)-by-\(n\) matrix but do not occur as the eigenvalues of a symmetric nonnegative \(n\)-by-\(n\)-matrix. This solves a problem posed by M. Boyle and D. Handelman [Trans. Am. Math. Soc. 336, No. 1, 121-149 (1993; Zbl 0766.15024)], D. Hershkowitz [Existence of matrices satisfying prescribed conditions. Thesis. Technion-Israel Inst. of Technology (1978)], and others. In the process, recent work by M. Boyle and D. Handelman [Ann. Math., II. Ser. 133, No. 2, 249-316 (1991; Zbl 0735.15005)] that solves the nonnegative inverse eigenvalue problem by appending 0’s to given spectral data is refined.
MSC:
| 15A18 | Eigenvalues, singular values, and eigenvectors |
| 15B48 | Positive matrices and their generalizations; cones of matrices |
References:
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