Matrices totally positive relative to a tree. (English) Zbl 1171.15021

Johnson, Charles R.; Costas-Santos, Roberto S.; Tadchiev, Boris

Electron. J. Linear Algebra 18, 211-221 (2009).

Summary: It is known that for a totally positive (TP) matrix, the eigenvalues are positive and distinct and the eigenvector associated with the smallest eigenvalue is totally nonzero and has an alternating sign pattern. Here, a certain weakening of the TP hypothesis is shown to yield a similar conclusion.

MSC:

15B48	Positive matrices and their generalizations; cones of matrices
15A18	Eigenvalues, singular values, and eigenvectors

Keywords:

totally positive matrices; Sylvester’s identity; graph theory; spectral theory; eigenvalues; eigenvector