Extended Abstracts of the 41 th Iranian International Conference on Mathematics 12-15 September 2010, University of Urmia, Urmia, Iran, pp 00-00 INNER TOTALLY POSITIVE MATRICES WITH TWO SPECTRUM IN COMMON K. GHANBARI 1* AND F. GILASI 1 Abstract. Let σ(A) be the set of eigenvalues of A and σ i (A) be the set of eigenvalues of Ai , where Ai is submatrix of A after deleting the i th rows i th column of A. Suppose ρ = {ρ1,ρ2,...,ρn} and γ = {γ1,γ2,...,γn} be staircase sequences. A matrix A ∈ Mn is called a staircase matrix with row and column, ρ, γ ; if a ij = 0 when i>γ(j ) or j>ρ(i). A minor 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 tatally positive(ITP ) if every inner minor of A is 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. This operations keep two spectrum of the given matrix, i.e. σ(A) and σ i (A). 1. Introduction and Preliminaries Let M n denote the set of real n × n matrices, and S n denote the subset of symmetric matrices. Let Q k,n denote the set of strictly increasing sequences of k integers α 1 ,α 2 ,...,α k taken from {1, 2,...,n}. We denote the submatrix of A with rows indexed by α = {α 1 ,...,α k } and columns indexed by β = {β 1 ,...,β k } 2000 Mathematics Subject Classification. 15A48; 15A23. Key words and phrases. Oscillatory matrices, totally positive matrices, inner totally positive matrices * Speaker. 1