I am reading an article by H. Minc: "Linear transformations on nonnegative matrices" (source https://core.ac.uk/download/pdf/82808273.pdf). There is a step in the proof of theorem 2 which I do not understand. Perhaps it is completely obvious but I just do not see it.
So what is what?
- $M_n(\mathbb C)$ is the set of all $n \times n$-matrices with elements in $\mathbb C$
- $E_{ij}$ is the $n \times n$-matrix with 1 at position $(i,j)$ and the value 0 everywhere else.
- $(S^{-1})^{(i)}$ is the $i$th column of matrix $S^{-1}$
- $S_{(j)}$ is the $j$th row of matrix $S$
- A nonnegtaive matrix has all entries $\geq 0$
..
Everything in the paper ist clear to me except the conclusion which is marked in red above. What am I missing? Thank you for your time and your thoughts!
The point is that clearly since $S$ is invertible, the inverse $S^{-1}$ is not the zero matrx, so we can find some $\lambda=(S^{-1})_{hi}\neq 0$. Then the previous inequality implies that for any $j,k$, we have $\lambda S_{jk}\geq 0$.
On the other hand, fix $j,k$. We can use the polar decomposition $\lambda=|\lambda|e^{\bf i\theta}$ and $S_{jk}=|S_{jk}|e^{\bf i\phi}$ and note that $\lambda S_{jk}=|\lambda| |S_{jk}|e^{\bf i(\phi+\theta)}\geq 0$, hence it must be that $\phi=-\theta \pmod{2\pi}$. In particular, setting $\alpha=E^{-\bf i \theta}$, we see that $S_{jk}=|S_{jk}|\alpha$ for all $j,k$.