# Encyclopedia of properties of nonnegative matrices

I'd like to buy a book that contains more or less all known properties of elementwise nonnegative nonnegative matrices, i.e. matrices $A$ such that $A_{i,j}\geq 0$ for all $i,j$.

After a bit googling I found these two books:

• It seems that Nonnegative Matrices in the Mathematical Sciences of Berman and Plemmons is exactly what I want. But it is more than 20 years old (1994), so I wonder if there is something more up-to-date.

• Nonnegative Matrices and Applications looks quite nice too and is a little bit more recent (1997). However, it seems that it is less oriented on the properties of nonnegative matrices themselves rather than their applications.

So, I'm a bit puzzled and my question is the following:

If I want to buy 1 book collecting the most complete list of known properties of nonnegative matrices, which one should I get?

• To avoid confusions (and it seems there have been some), I'd recommend making it explicit that you are after elementwise nonnegative matrices. – Suvrit Dec 10 '15 at 19:16
• Bapat's book is pretty much the closest to what you are seeking. I am not aware of another book that matches your requirements. Hence my answer is: $\emptyset$ – Suvrit Dec 10 '15 at 22:20

## 2 Answers

In addition to the books you list, there is also the book Nonnegative Matrices by H. Minc (1974), but it is outdated and out of print.

Also, the book Combinatorial Matrix Theory by Brualdi and Ryser does a nice treatment of the combinatorial structure of nonnegative matrices (and complex matrices in general).

IMHO, a new textbook on nonnegative matrices is long overdue (an entire monograph could be devoted to the nonnegative inverse eigenvalue problem alone).

Positive matrices are only one side of a more general subject, that of Totally positive matrices. So you can be interested in the books written by Allan Pinkus (Cambridge Univ. Press, 2010) or by Shaun Fallat & Charles Johnson (Princeton niv. Press, 2011).

• Thank you for your answer. I'm not sure to understand how Positive matrices are only one side of [...] Totally positive matrices. It seems to me that $A=\begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix}$ is a positive matrix which is not totally positive (its determinant is strictly negative). – Surb Dec 10 '15 at 14:25
• @Surb. Precisely, that's the point. Total positivity means that $A$ is positive, and it induces positive matrices when acting on the exterior algebra. This implies that, not only $A$ has a positive real eigenvalue, but all its eigenvalues are positive (the converse is false). – Denis Serre Dec 10 '15 at 16:32
• As I understand this the "totally positive matrix" are a subset of the OP's nonnegative matrices. So I think the qualification in this answer "...of a more general subject,..." is a bit misleading. – Gottfried Helms Dec 10 '15 at 21:55
• I agree with @Gottfried Helms, the set of totally positive matrices seems to be a subset of the set of entrywise nonnegative matrices. Thus I don't see why > positive matrices are just one side of a more general subject, that of Totally positive matrices. At the same time I don't see the relation between entrywise nonnegative (positive) matrices and nonnegative (or positive) eigenvalues, neither their real part (except, of course, for the spectral radius). For instance, consider the matrix $E$ made by all ones. (cont.) – FFT Dec 11 '15 at 11:43
• The matrix $A = E -\epsilon I$, is a positive matrix for any $0<\epsilon <1$ and its eigenvalues are $n-\epsilon >0$ (simple Perron eigenvalue) and $-\epsilon$ with multiplicity $n-1$. So $A$ has $n-1$ negative eigenvalues. – FFT Dec 11 '15 at 11:43