Frobenius-Perron dimension on a fusion category

Question

Let $C$ be a fusion category with simple objects $V_i\in I$. The fusion rule is $V_i\otimes V_j \cong N_{i,j}^k V_k$.

The Frobenius-Perron dimension of a simple object $V_i$, $\mathrm{FPdim}(i)$, is defined as the largest nonnegative real eigenvalue of a matrix $N_i$, where the $(k,j)$ entry is $(N_i)_{k j}=N_{i,j}^k$.

I would like to know if we have the following equality or not.

$\sum_{j\in I} \mathrm{FPdim}(j) N_{i j}^k=\mathrm{FPdim}(i) \mathrm{FPdim}(k)$

for any fixed $i, k \in I$.

If so, I would like to know how to prove this. Thank you.