Abstract :
Suppose Aset membership, variantMn is a staircase matrix with row and column staircase sequences ρ={ρ(1),ρ(2),…,ρ(n)}, γ={γ(1),γ(2),…,γ(n)}. A minor A(α;β)=det(A[αβ]) with α={α1…,αk},β={β1,…,βk} is said to be an inner minor of A if αiless-than-or-equals, slantγ(βi),βiless-than-or-equals, slantρ(αi) for i=1,2,…,k. A is said to be inner totally positive (ITP) if every inner minor of A is positive. We prove that A is ITP if A(α;β)>0 for all inner minors with α,βset membership, variantQk,n0,k=1,2,…,n. Also Aset membership, variantMn is ITP iff it is totally non-negative and its extreme inner minors are positive. We show that an ITP matrix may be reduced by similarity transformations to an ITP band matrix, and may alternatively be filled-in by similarity transformations to become a TP matrix. In both the reduction and the filling in, the matrix is ITP at each stage. The analysis is applied to some inverse eigenvalue problems for band matrices.
