Matrices whose inverse is positive - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T22:25:41Z http://mathoverflow.net/feeds/question/91515 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/91515/matrices-whose-inverse-is-positive Matrices whose inverse is positive hydrobates 2012-03-18T08:57:00Z 2012-05-25T00:11:40Z <p>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.</p> http://mathoverflow.net/questions/91515/matrices-whose-inverse-is-positive/91517#91517 Answer by Anatoly Kochubei for Matrices whose inverse is positive Anatoly Kochubei 2012-03-18T09:33:21Z 2012-03-18T09:33:21Z <p>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, <a href="http://ac.els-cdn.com/002437959300244T/1-s2.0-002437959300244T-main.pdf?_tid=9e7b73aff99f6a5e36bde4b1265a396c&amp;acdnat=1332062723_3c39ada8e0b1d67ca7f0aedf2034ae45" rel="nofollow">http://ac.els-cdn.com/002437959300244T/1-s2.0-002437959300244T-main.pdf?_tid=9e7b73aff99f6a5e36bde4b1265a396c&amp;acdnat=1332062723_3c39ada8e0b1d67ca7f0aedf2034ae45</a></p> <p>or</p> <p><a href="http://www.math.temple.edu/~abed/JS07.pdf" rel="nofollow">http://www.math.temple.edu/~abed/JS07.pdf</a></p> <p>In these papers (and those referred to by them) you will find methods to construct other classes of matrices with this property.</p> http://mathoverflow.net/questions/91515/matrices-whose-inverse-is-positive/91532#91532 Answer by David Speyer for Matrices whose inverse is positive David Speyer 2012-03-18T14:08:46Z 2012-03-18T14:08:46Z <p>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 $(-1)^{n+1} (-1)^{i-j} \det A_{ij} > 0$. </p> <p>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.</p> <p>In particular, we have an explicit formula for $A^{-1}$. The entry $(A^{-1})_{ij}$ is <code>$$(-1)^{i-j} \frac{b_{j+1} b_{j+2} \cdots b_{i-1} c_{i+1} c_{i+2} \cdots c_j}{\det A}.$$</code></p> http://mathoverflow.net/questions/91515/matrices-whose-inverse-is-positive/97883#97883 Answer by Felix Goldberg for Matrices whose inverse is positive Felix Goldberg 2012-05-24T23:55:42Z 2012-05-25T00:11:40Z <p>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:</p> <p><a href="http://books.google.ie/books/about/Nonnegative_Matrices_in_the_Mathematical.html?id=MRB7SUc_u6YC&amp;redir_esc=y" rel="nofollow">http://books.google.ie/books/about/Nonnegative_Matrices_in_the_Mathematical.html?id=MRB7SUc_u6YC&amp;redir_esc=y</a></p> <p>In the case of this question, the matrix might not be an $M$-matrix. It depends on the actual entries.</p>