Matrices totally positive relative to a tree. II. (English) Zbl 1337.15025

Costas-Santos, R. S.; Johnson, C. R.

Linear Algebra Appl. 505, 1-10 (2016).

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

Keywords:

graph; Neumaier conclusion; spectral theory; Sylvester’s identity; totally positive matrix; totally positive relative to a tree

Citations:

Zbl 1171.15021

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 University Press Princeton, NJ · Zbl 1390.15001
[3]	Horn, R. A.; Johnson, C. R., Matrix Analysis (2013), Cambridge University Press: Cambridge University Press Cambridge, xviii+643 pp · Zbl 1267.15001