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A note on the inverse eigenvalue problem for symmetric doubly stochastic matrices. (English) Zbl 1193.15009

S.-G. Hwang and S.-S. Pyo [ibid. 379, 77–83 (2004; Zbl 1040.15010)] had claimed as a proposition that a real \(n\)-tuple \(1\geq \lambda_2 \geq \cdots \geq \lambda_n\) such that
\[ \frac{1}{n} +\frac{\lambda_2}{n(n-1)}+\frac{\lambda_3}{(n-1)(n-2)}+\ldots +\frac{\lambda_n}{2(1)}\geq 0 \]
is the spectrum of a symmetric doubly stochastic matrix. In this note, the author points out by a counterexample that the proposition is wrong.

MSC:

15A18 Eigenvalues, singular values, and eigenvectors
15A29 Inverse problems in linear algebra
15B51 Stochastic matrices

Citations:

Zbl 1040.15010
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References:

[1] Horn, R. A.; Johnson, C. R., Matrix Analysis (1985), Cambridge University Press · Zbl 0576.15001
[2] Hwang, S. G.; Pyo, S. S., The inverse eigenvalue problem for symmetric doubly stochastic matrices, Linear Algebra Appl., 379, 77-83 (2004) · Zbl 1040.15010
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