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 $R^n$. 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, how can I characterize the above defined matrices? 1 # 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)? 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$R^n$. 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$x$if its Jacobian matrix evaluated at$x$, which we label as$A\$, satisfies the following condition:
$$v'Av\geq 0,\quad \forall v \text{ such that } 1'v=0.$$