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Matrices totally positive relative to a tree. II. (English) Zbl 1337.15025
Summary: If \(T\) is a labelled tree, a matrix \(A\) is totally positive relative to \(T\), principal submatrices of \(A\) associated with deletion of pendent vertices of \(T\) are P-matrices, and \(A\) has positive determinant, then the smallest absolute eigenvalue of \(A\) is positive with multiplicity 1 and its eigenvector is signed according to \(T\). This conclusion has been incorrectly conjectured under weaker hypotheses.
For Part I see [C. R. Johnson et al., Electron. J. Linear Algebra 18, 211–221 (2009; Zbl 1171.15021)].
MSC:
15B48 Positive matrices and their generalizations; cones of matrices
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
15A18 Eigenvalues, singular values, and eigenvectors
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References:
[1] Johnson, C. R.; Costas-Santos, R. S.; Tadchiev, B., Matrices totally positive relative to a tree, Electron. J. Linear Algebra, 18, 211-221, (2009) · Zbl 1171.15021
[2] Fallat, S.; Johnson, C. R., Totally nonnegative matrices, Princeton Series in Applied Mathematics, (2011), Princeton University Press Princeton, NJ · Zbl 1390.15001
[3] Horn, R. A.; Johnson, C. R., Matrix analysis, (2013), Cambridge University Press Cambridge, xviii+643 pp · Zbl 1267.15001
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