Bidiagonal Decompositions of Oscillating Systems of Vectors
نویسندگان
چکیده
We establish necessary and sufficient conditions, in the language of bidiagonal decompositions, for a matrix V to be an eigenvector matrix of a totally positive matrix. Namely, this is the case if and only if V and V −T are lowerly totally positive. These conditions translate into easy positivity requirements on the parameters in the bidiagonal decompositions of V and V −T . Using these decompositions we give elementary proofs of the oscillating properties of V . In particular, the fact that the jth column of V has j − 1 changes of sign. Our new results include the fact that the Q matrix in a QR decomposition of a totally positive matrix belongs to the above class (and thus has the same oscillating properties).
منابع مشابه
Ela Bidiagonal Decompositions, Minors and Applications
Abstract. Matrices, called ε-BD matrices, that have a bidiagonal decomposition satisfying some sign constraints are analyzed. The ε-BD matrices include all nonsingular totally positive matrices, as well as their matrices opposite in sign and their inverses. The signs of minors of ε-BD matrices are analyzed. The zero patterns of ε-BD matrices and their triangular factors are studied and applied ...
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