Exercise 5.6.2

If $A$ has all positive entries, then $A^{T}A$ and $A{A}^T$ have all positive entries. let $A=U\mathrm{\Sigma }{V}^T$, we have $A^{T}A=V{\mathrm{\Sigma }}^2{V}^T=V{\mathrm{\Sigma }}^2{V}^{-1}$ and $A{A}^T=U{\mathrm{\Sigma }}^2{U}^T=U{\mathrm{\Sigma }}^2{U}^{-1}$. So $V$ and $U$ have the eigenvectors of $A^{T}A$ and $A{A}^T$ respectively. By Perron’s theorem, we have $v_1>0$ and $u_1>0$. Also by definition of SVD, the singular values of $A$ are positive. So we see that the rank 1 matrix ${\sigma }_1{u}_1{v}_1^T$ is also positive.