Question

I have recently come across some examples of matrices with a special structure. I will describe these matrices here and I hope that somebody will be able to point out a source where I can find more information about them. Consider an $n\times n$ matrix $A$ with elements $a_{ij}$ having the following properties. The elements with $i=j$ (call them $b_i$) are negative. The elements with $j=i+1 {\rm mod} n$ (call then $c_i$) are positive. All other elements are zero. The determinant of a matrix of this type is $\sum_i b_i+(-1)^{n+1}\sum_i c_i$. A property of these matrices which I found surprising is that $(-1)^{n+1}(\det A)A^{-1}$ is a positive matrix, i.e. all its entries are positive. I found this by playing around with some examples. Can anybody point out to me some general theory which explains this observation? I met these matrices repeatedly when looking at certain chemical reaction networks. In that context the positivity statement is valuable because it allows the Perron-Frobenius theorem to be applied.

Answer 1

A class of matrices with entrywise positive inverses (inverse-positive matrices) appears in a variety of applications and has been studied by many authors. See, for example, http://ac.els-cdn.com/002437959300244T/1-s2.0-002437959300244T-main.pdf?_tid=9e7b73aff99f6a5e36bde4b1265a396c&acdnat=1332062723_3c39ada8e0b1d67ca7f0aedf2034ae45

or

http://www.math.temple.edu/~abed/JS07.pdf

In these papers (and those referred to by them) you will find methods to construct other classes of matrices with this property.

Answer 2

This is straightforward from the adjoint formula for the inverse matrix. Let $A_{ij}$ be the matrix formed by deleting row $i$ and column $j$ from $A$. We must show that $(-1)^{n+1} (-1)^{i-j} \det A_{ij} > 0$.

We can reorder the rows and columns of $A$ cyclically to assume without loss of generality that $j=n$. Then $A_{in}$ is block diagonal with two blocks of size $i-1$ and $n-i$. The first block is upper diagonal with diagonal entries $b_1 b_2 \cdots b_{i-1}$; the second block is lower diagonal with diagonal entries $c_{i+1} c_{i+2} \cdots c_n$. So $\det A_{ij}$ has sign $(-1)^{i-1} = (-1)^{n+1} (-1)^{n-i}$ as desired.

In particular, we have an explicit formula for $A^{-1}$. The entry $(A^{-1})_{ij}$ is $$ (-1)^{i-j} \frac{b_{j+1} b_{j+2} \cdots b_{i-1} c_{i+1} c_{i+2} \cdots c_j}{\det A}.$$

Answer 3

Matrices whose inverses are nonnegative are also called monotone. There are a number of equivalent characterizations in Theorem 6.2.3 of the wonderful book by Berman and Plemmons:

http://books.google.ie/books/about/Nonnegative_Matrices_in_the_Mathematical.html?id=MRB7SUc_u6YC&redir_esc=y

In the case of this question, the matrix might not be an $M$-matrix. It depends on the actual entries.