Background:
Let $x_1,\ldots,x_n$ be the variables satisfying the equations of motion $\ddot{x_i}=f_i(x_1,\ldots,x_n)$ for $i=1,\ldots,n$
We introduce a small perturbation such that $x_i(t)=x_i^0 +h_i(t)$ where $x_i^0$ is the equilibrium point.
Then in matrix form $\ddot{\vec{h}}=A\vec{h}$ where $A_{ij}=\left({\partial f_i\over \partial x_j}\right)$
Now, it is said that $A$ in general has different right $w_k$ and left $v_k$ eigenvectors such that $Aw_k=\lambda_k w_k$ and $v_k^TA=\lambda_kv_k^T$.
Why is it true that $v_a\cdot w_b=\delta_{ab}$?
If it is at all relevant, we are given that the system has only real eigenvalues.
Thank you.