On the Matrix Equation X ′ X = A

Source 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.