On generators of bounded ratios of minors for totally positive matrices
نویسندگان
چکیده
We provide a method for factoring all bounded ratios of the form det A(I1|I ′ 1) det A(I2|I ′ 2)/ det A(J1|J ′ 1) det A(J2|J ′ 2) where A is a totally positive matrix, into a product of more elementary ratios each of which is bounded by 1, thus giving a new proof of Skandera’s result. The approach we use generalizes the one employed by Fallat et al. in their work on principal minors. We also obtain a new necessary condition for a ratio to be bounded for the case of non-principal minors.
منابع مشابه
Bounded Ratios of Products of Principal Minors of Positive Definite Matrices
Considered is the multiplicative semigroup of ratios of products of principal minors bounded over all positive definite matrices. A long history of literature identifies various elements of this semigroup, all of which lie in a subsemigroup generated by Hadamard-Fischer inequalities. Via cone-theoretic techniques and the patterns of nullity among positive semidefinite matrices, a semigroup cont...
متن کاملMultiplicative Principal-minor Inequalities for a Class of Oscillatory Matrices
A square matrix is said to be totally nonnegative (respectively, positive) if all of its minors are nonnegative (respectively, positive). Determinantal inequalities have been a popular and important subject, especially for positivity classes of matrices such as: positive semidefinite matrices, M−matrices, and totally nonnegative matrices. Our main interest lies in characterizing all of the ineq...
متن کاملTotal Positivity in Loop Groups I: Whirls and Curls
This is the first of a series of papers where we develop a theory of total positivity for loop groups. In this paper, we completely describe the totally nonnegative part of the polynomial loop group GLn(R[t, t ]), and for the formal loop group GLn(R((t))) we describe the totally nonnegative points which are not totally positive. Furthermore, we make the connection with networks on the cylinder....
متن کاملEla Zero Minors of Totally Positive Matrices
In this paper the structure of the zero minors of totally positive matrices is studied and applications are presented.
متن کاملEfficient Recognition of Totally Nonnegative Matrix Cells
The space of m × p totally nonnegative real matrices has a stratification into totally nonnegative cells. The largest such cell is the space of totally positive matrices. There is a well-known criterion due to Gasca and Peña for testing a real matrix for total positivity. This criterion involves testing mp minors. In contrast, there is no known small set of minors for testing for total nonnegat...
متن کامل