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Notations:

Let $M = (m_{ij})\in \mathbb R^{n\times n}$ be a square real matrix, such that $m_{ij}\leq 0$ for all $i\neq j$. We say that a matrix is monotone when $M$ is invertible and $M^{-1}$ has all its coefficients positive. We call "leading principal minors" the $\det (m_{ij})_{1\leq i,j\leq k}$ for $1\leq k \leq n$.

The problem:

Show that $M$ is monotone $\iff$ all the leading principal minors of $M$ are $>0$.

What I've done so far:

I'm really stuck with this one. Tried lots of things but nothing seems to work yet.

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  • $\begingroup$ Yes, you are right. I fixed it. However, I didn't find anything on the internet to show this equivalence simply (without showing a lot of other equivalence). Do you have any idea? $\endgroup$
    – MiKiDe
    May 18 '20 at 17:27
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These are actually two equivalent characterisations of nonsingular $M$-matrices. In standard terminologies for nonnegative matrices, it can be stated as follows: let $M\in\mathbb R^{n\times n}$ be a nonsingular $Z$-matrix. Then $M$ is inverse-positive if and only if all leading principal minors of $M$ are positive.

A proof of $50$ equivalent characterisations of nonsingular $M$-matrices can be found in theorem 2.3, chapter 6 of Nonnegative Matrices in the Mathematical Sciences written by Berman and Plemmons. The equivalence of the two particular characterisations in your question can be proved as follows.

Suppose $M$ is inverse-positive. Let $\alpha=\max_im_{ii}$. Then $P=\alpha I-M\ge0$. Let $x\ge0$ be a Perron vector for $P$. Then $(\alpha-\rho(P))M^{-1}x=x\ge0$. Since $M^{-1}x$ is also nonnegative, we must have $\alpha>\rho(P)$. Now denote by $P_k$ the leading principal $k\times k$ submatrix of $P$ and define $M_k$ analogously. Since $P\ge0$, by using Gelfand's formula with the induced $\infty$-norm, we have $\rho(P_k)\le\rho(P)$. It follows that $\alpha-\rho(P_k)\ge\alpha-\rho(P)>0$. Consequently, all eigenvalues of $M_k=\alpha I_k-P_k$ lie on the open right half plane. Therefore $\det(M_k)>0$ for each $k$, i.e. $M$ has positive leading principal minors.

Conversely, suppose $M$ has positive leading principal minors. By mathematical induction on $n$, one can show that $M$ has an LU-decomposition $M=LU$ such that all diagonal elements of $L$ are equal to $1$ and $L,U$ are inverse-positive. More specifically, in the inductive step, we have \begin{aligned} \pmatrix{L_0U_0&x\\ y^T&a} &=\pmatrix{L_0&0\\ y^TU_0^{-1}&1}\pmatrix{U_0&L_0^{-1}x\\ 0&b} \quad(b=a-y^TU_0^{-1}L_0^{-1}x),\\ \pmatrix{L_0&0\\ y^TU_0^{-1}&1}^{-1}&=\pmatrix{L_0^{-1}&0\\ -y^TU_0^{-1}L_0^{-1}&1},\\ \pmatrix{U_0&L_0^{-1}x\\ 0&b}^{-1}&=\pmatrix{U_0^{-1}&-b^{-1}U_0^{-1}L_0^{-1}x\\ 0&b^{-1}}. \end{aligned} It follows that $M^{-1}=U^{-1}L^{-1}$ is nonnegative, i.e. $M$ is inverse-positive.

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