Abstract.In this paper it is shown that, given a complex square matrix A all of whose leading principal minors are nonzero, there is a diagonal matrix D such that the product DA of the two matrices has all its characteristic roots positive and simple. This result is already known for real A, but two new proofs for this case are given here. We shall give here two proofs of Theorem 1, both of them simpler than the proof in [2]. Our first proof is the shorter of the two, but is less constructive since it makes use of the continuity of the roots (as functions of the matrix entries). Our second proof gives explicit (and relatively simple) estimates for the entries of D in terms of the entries of A.First proof of Theorem 1. Here we use induction on n. For « = 1 the result is trivial, so suppose that n 5: 2 and that the result holds for matrices of order n -1. Let A be an «X« real matrix all of whose lpm's (leading principal minors) are positive and let ^4i be its leading principal submatrix of order n-1. Then all the lpm's of A\ are positive, so by our induction assertion there is a positive diagonal matrix D\ of order n-1 such that all roots of D1A1 are positive and simple. Let d be a real number to be determined later (but treated as a variable for the present). Let A be partitioned as follows:U3 Ad
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