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Row stochastic inverse eigenvalue problem. (English) Zbl 1269.15008

Summary: We give sufficient conditions or realizability criteria for the existence of a row stochastic matrix with a given spectrum \(\Lambda = \{\lambda_1, \dots, \lambda_n\} = \Lambda_1 \cup \ldots \cup \Lambda_m \cup \Lambda_{m+1}\), \(m > 0\); where \(\Lambda_k=\{\lambda_{k1}, \lambda_{k2},\dots, \lambda_{kp_k}\}= \{\lambda_{k1}, \omega_{k}e^{2\pi i/p_k}, \omega_ke^{4\pi i/p_k},\dots, \omega_ke^{2(p_k-1)\pi i/p_k}\}\) (\(p_k\) is an integer greater than 1), \(\lambda_{k1}=\lambda_k > 0\), \(1 = \lambda_1 \geq \omega_k > 0\), \(k =1, \dots, m\); \(\Lambda_{m+1} = \{\lambda_m+1\}\), \(\omega_ {m+1} \equiv \lambda_1 + \dots + \lambda_n \leq \lambda_1\), \(\omega_k \geq \lambda_k\), \(\omega_1 \geq \lambda_k\), \(k= 2, \ldots, m+1\). In the case when \(p_1, \dots, p_m \) are all equal to \(2\), \(\Lambda\) becomes a list of \(2m + 1\) real numbers for any positive integer \(m\), and our result gives sufficient conditions for a list of \(2m + 1\) real numbers to be realizable by a row stochastic matrix.

MSC:

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

[1] doi:10.1016/j.laa.2005.12.026 · Zbl 1097.15014
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