Eigenvalues of a matrix with all diagonal elements zero

Question

Hi I want to check if the following matrix has any eigenvalues with positive real part $$ \begin{pmatrix} 0 & \displaystyle\frac{b_3 (J_3-J_2) k_p}{J_1} & \displaystyle\frac{b_2 (J_3-J_2) k_p}{J_1} \\ \displaystyle\frac{b_3 (J_1-J_3) k_p}{J_2} & 0 & \displaystyle\frac{b_1 (J_1-J_3) k_p}{J_2} \\ \displaystyle\frac{b_2 (J_2-J_1) k_p}{J_3} & \displaystyle\frac{b_1 (J_2-J_1) k_p}{J_3} & 0 \\ \end{pmatrix} $$ Note that $||b||=1$ and $J_1>0$, $J_2>0$ and $J_3>0$.

PS: determinant of the matrix is $\dfrac{-2*b_1*b_2*b_3*(J_1-J_2)(J_3-J_1)(J_2-J_3)k_p^3}{J_1J_2J_3}$

Answer 1

The matrix does not always possess an eigenvalue with a positive real part. E.g. it is nilpotent when $b_3=0$ and $J_1=J_2$.