## 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

Zbl 1171.15021
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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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