A new proof for \(DP_n=CP_n\) \((n\geq 4)\)
Let \(A\) be an \(n\times n\) real symmetric matrix. Then \(A\) is called completely positive \((\in CP_n)\) if \(A\) can be factored as \(A= BB^T\) for some real \(n\times m\) matrix \(B\) with nonnegative entries. The smallest \(m\) is denoted by \(\phi(A)\) and clearly \(\phi(A)\geq\text{rank}(A)\). A matrix is called doubly nonnegative \((\in DP_n)\) if it has nonnegative entries and is positive semidefinite. The following results are well-known: (a) \(DP_n= CP_n\) for \(n\leq 4\), (b) \(CP_n\subset DP_n\) for \(n\geq 5\). The proofs of (a) employ a variety of methods and are not simple. In this paper the authors give a more direct, matrix-theoretical proof of (a). They also show that \(\phi(A)\leq n\) for \(n\leq 4\).
scientific article; zbMATH DE number 5142502