A matrix $A$ is called conditionally positive definite (or CPD) if $\langle x,Ax\rangle \geq 0$ for all $x \in \mathbb{R}^n$ such that $x_1+\cdots+x_n=0$, see p. 180 of below book. The matrix $A$ is called positive definite if $\langle x,Ax\rangle > 0$ for all $0\neq x \in \mathbb{R}^n$. The matrix $A$ is called positive semidefinite if $\langle x,Ax\rangle \geq 0$ for all $x \in \mathbb{R}^n$.
There are two theorems given in the book "Positive definite matrices, by R. Bhatia".
1). (Exe 5.6.16, p. 182) Let $A$ be a symmetric matrix of order $n$. Let $D$ be a matrix of order $(n-1)$ given by $$d_{ij}=a_{ij}+a_{i+1,j+1}-a_{i,j+1}-a_{i+1,j}.$$ Then $A$ is CPD if $D$ is positive definite.
2). (Exe 5.6.14, p. 180) If $A$ is CPD then $e^A=[e^{a_{ij}}]$ is positive definite.
I am confused with the sign $\geq$ in the definition of CPD. This leads me to think if there should be "positive semidefinite" instead of "positive definite" in the above two results. Actually, I have a matrix $A$ for which the matrix $D$ is positive definite, so using the above two theorems, I concluded that $e^{A}$ is positive definite. Am I correct?
Any suggestion or help is really appreciated!