On the Matrix Equation X′X = A

Abstract

If X is a matrix with non-negative entries then X′X is positive semi-definite with non-negative entries. Conversely, if A is positive semi-definite then there exist matrices Y, not necessarily with non-negative entries, such that Y′Y = A. In the present paper we investigate whether, given a positive semidefinite matrix A with non-negative entries, the equation X′X = A has a solution X with non-negative entries. An equivalent statement of the problem is: Can a positive semi-definite matrix with non-negative entries be expressed as a sum of rank 1 positive semi-definite matrices with non-negative entries? We answer the question in the affirmative for n≦4 and quote the following example due to M.

Unbounded convex sets for non-convex mixed-integer quadratic programming

“…Proposition 8 states that MIQ 0,n is completely described by psd inequalities, for all n. On the other hand, MIQ + 0,n is completely described by psd and non-negativity inequalities if and only if n ≤ 3. (This follows from Proposition 7, together with the fact, from [25], that the set of completely positive matrices is equal to the set of doubly non-negative matrices if and only if n ≤ 4.) In particular, one sees that MIQ 0,1 is also described by the single convex quadratic inequality y 11 ≥ x 2 1 , and that MIQ + 0,1 is described by the convex quadratic inequality y 11 ≥ x 2 1 and the non-negativity inequality…”

Copositive Programming – a Survey

“…It is a very interesting fact (cf. [41]) that for n × n-matrices of order n ≤ 4, we have equality in the above relations, whereas for n ≥ 5, both inclusions are strict. A counterexample that illustrates C = S + + N is the so-called Horn-matrix, cf.…”

First order solutions in conic programming

“…Example 3.7 [COP with n = m = 2] It is known that COP 2 = N 2 + S + 2 and (COP 2 ) * = CP 2 = N 2 ∩ S + 2 , see Maxfield and Minc (1962)…”