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Matrices Totally Positive Relative to a Tree, II

Author:
Costas-Santos, Roberto S.; Johnson, Charles R.
URI:
https://hdl.handle.net/20.500.12412/6554
DOI:
10.1016/j.laa.2016.04.021
Date:
2016-09-15
Keyword(s):

Graph

Neumaier conclusion

Spectral theory

Sylvester’s identity

Totally positive matrix

Totally positive relative to a tree

Abstract:

If T is a labelled tree, 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.

If T is a labelled tree, 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.

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Attribution-NonCommercial-NoDerivatives 4.0 Internacional
Except where otherwise noted, this item's license is described as Attribution-NonCommercial-NoDerivatives 4.0 Internacional