# Square root of doubly positive symmetric matrices

I wonder whether the following property holds true: For every real symmetric matrix $$S$$, which is positive in both senses: $$\forall x\in{\mathbb R}^n,\,x^TSx\ge0,\qquad\forall 1\le i,j\le n,\,s_{ij}\ge0,$$ then $$\sqrt S$$ (the unique square root among positive semi-definite symmetric matrices) is positive in both senses too. In other words, it is entrywise non-negative.

At least, this is true if $$n=2$$. By continuity of $$S\mapsto\sqrt S$$, we may assume that $$S$$ is positive definite. Denoting $$\sqrt S=\begin{pmatrix} a & b \\ b & c \end{pmatrix},$$ we do have $$a,c>0$$. Because $$s_{12}=b(a+c)$$ is $$\ge0$$, we infer $$b\ge0$$.

## 2 Answers

No. If $$A = \begin{pmatrix}10&-1&5\\-1&10&5\\5&5&10\end{pmatrix},$$ then $$A$$ is positive definite but does not have all entries positive, while $$A^2 = \begin{pmatrix}126&5&95\\5&126&95\\95&95&150\end{pmatrix}$$ is positive in both senses.

Robert Bryant already showed via an example that the answer is "no". To come up with lots of counterexamples, recall that (under some mild assumptions) if $$A$$ has maximal eigenvalue $$\lambda_{\text{max}}$$ and corresponding unit eigenvector $$\mathbf{v}$$ then $$(A/\lambda_{\text{max}})^k \rightarrow \mathbf{v}\mathbf{v}^*$$ as $$k \rightarrow \infty$$.

So if you pick any matrix that is (a) positive semidefinite with a negative entry, and (b) has a unique maximal eigenvalue with corresponding entrywise positive eigenvector, then repeatedly squaring $$A$$ will eventually give a counterexample to the original question.

• I like this conceptual analysis; it's what I had in mind when I constructed the example that I did. In order to really complete this as an argument, though, one needs to explain that, while there is no symmetric $n$-by-$n$ matrix $A$ that satisfies both conditions (a) and (b) when $n=2$, there is such a matrix when $n=3$ (and hence, when $n\ge 3$). Of course, once one has produced a counterexample when $n=3$, there's no need for the general argument. Oct 19 '20 at 11:48