I derive this question while trying to prove the monotonicity of a differentiable vector function $f(x)$ that maps from $X\subset R^n$ to $R^n$ (Here function $f(x)$ is called monotone if $(x-y)'(f(x)-f(y))\geq 0$, $\forall x,y\in X$). The domain $X$ only consists of vectors $x$ such that $1'x=0$, here $1$ is the vector of all ones.


Using the mean-value theorem, we have that $f(x)$ is locally monotone at $x$ (namely $(y-x)'(f(y)-f(x))\geq 0$, $\forall y\in X$) if its Jacobian matrix evaluated at $x$, which we label as $A$, satisfies the following condition:


$$v'Av\geq 0,\quad \forall v \text{ such that } 1'v=0.$$


This is a weaker condition than positive semidefiniteness. However, while there are a number of ways to characterize positive semidefinite matrices, for example, see [this Wikipedia page](http://en.wikipedia.org/wiki/Positive-semidefinite_matrix#Characterizations), how can I characterize the above defined matrices?



How to characterize real square matrices A, such that v'Av >= 0, for all real vectors v with 1'v=0 (1 is the vector of all ones)?
If $A$ is symmetric, then the matrices that you mention are called:


**Conditionally positive definite** (CPD) --- these are intimately related to the venerable *infinitely divisible matrices*


There is a vast amount of literature on these matrices, some useful pointers can already be found in R. Bhatia's wonderful book: *Positive definite matrices*


There are some basic algorithmic approaches to check whether a matrix is CPD or not (e.g., Ref. 3 below)


A simple characterization is given by the following. Let $A$ be an $n \times n$ Hermitian matrix, and let $B$ be the $(n-1) \times (n-1)$ matrix with entries


$$b\_{ij} = a\_{ij} + a\_{i+1,j+1} - a\_{i,j+1} - a\_{i+1,j}$$


Then $A$ is CPD *if and only if* $B$ is positive-definite.


**References**


1. R. Bhatia. *Positive definite matrices* (Chapter 5)
2. R. B. Bapat and T. E. S. Raghavan. Nonnegative matrices and applications (Chapter 4)
3. Kh. D. Ikramov and N. V. Savel'eva. *Conditionally positive definite matrices*, J. Mathematical Sciences, Vo. 98, No. 1, 2000.
4. R. A. Horn. The theory of infinitely divisible matrices and kernels (e.g. here : <http://www.ams.org/journals/tran/1969-136-00/S0002-9947-1969-0264736-5/S0002-9947-1969-0264736-5.pdf>)


