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Some lower bounds for the spectral radius of matrices using traces. (English) Zbl 1197.15010

Authors’ abstract: Let \(A\) be an \(n \times n\) matrix with eigenvalues \(\lambda_1, \lambda_2, \dots, \lambda_n\), and let \(m\) be an integer satisfying rank\((A) \leq m \leq n\). If \(A\) is real, the best possible lower bound for its spectral radius in terms of \(m\), \(\text{tr }A\) and \(\text{tr }A^2\) is obtained. If \(A\) is any complex matrix, two lower bounds for \(\sum_{j=1}^n |\lambda_j|^2\) are compared, and furthermore a new lower bound for the spectral radius is given only in terms of \(\text{tr }A\), \(\text{tr }A^2\), \(\|A\|\), \(\|A^*A - AA^*\|\), \(n\) and \(m\).

MSC:

15A42 Inequalities involving eigenvalues and eigenvectors
15A15 Determinants, permanents, traces, other special matrix functions
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References:

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