Positive symmetric matrices with exactly one positive eigenvalue (Q1002259)

Let~\(\mathcal A\) denote the set of positive and symmetric matrices with exactly one positive eigenvalue. A real and symmetric~\(n\times n\) matrix~\(A\) is called conditionally negative definite~(cnd) if~\(x^TAx\leq 0\) for all \(x=(x_i)\in\mathbb{R}^n\) such that \(x_1+\cdots +x_n=0\). If~\(A\) is positive and cnd, then~\(A\in\mathcal A\) [see Corollary~4.1.5 of~\textit{R.~B.~Bapat} and \textit{T.~E.~S.~Raghavan}, Nonnegative matrices and applications. Encyclopedia of Mathematics and Its Applications. 64. Cambridge: Cambridge University Press (1997; Zbl 0879.15015)]. If and only if~\(A\in\mathcal A\), then the (unique) doubly stochastic matrix~\(D^TAD\) where~\(D\) is a diagonal matrix with positive diagonal entries is cnd, see Theorem~4.4.6 of Bapat and Raghavan [op. cit.] Let~\(A\) be positive and nonsingular. The author proves that~\(A\in\mathcal A\) if and only if~\(A=LDL^T\) where \(L\) is a unit lower triangular matrix and \(D\) is a nonsingular diagonal matrix with the first diagonal entry positive and all remaining diagonal entries negative. He also presents three necessary or sufficient conditions for~\(A\in\mathcal A\). Finally, he studies pivoting strategies with minimal growth factor, when Gaussian elimination is applied to~\(A=(a_{ij})\in\mathcal A\). The growth factor with a given pivoting stategy is defined by \[ \rho_n(A)=\frac{\max_{i,j,k}|a_{ij}^{(k)}|}{\max_{i,j}|a_{ij}|} \] where~\(a_{ij}^{(k)}\) is the~\((i,j)\)'th entry after the \(k\)'th elimination. \{Reviewer's remark: On p.~1567, row~3, the reference~(10) should read~(18).\}