Suppose A∈ M n 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[ α| β]) with α={ α 1…, α k }, β={ β 1,…, β k } is said to be an inner minor of A if α i ⩽ γ( β i ), β i ⩽ ρ( α i ) 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, β∈ Q k, n 0, k=1,2,…, n. Also A∈ M n 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.