Show that using Suffix Notation

Question

I'm not sure what this would be in Suffix notation: $$\mathbf{Tr}\left[\left(\underline{\nabla}\,\underline{u}\right)^2\right]$$ I've got that: $$\left[\underline{\nabla}\,\underline{u}\right]_i=\frac{\partial u_i}{\partial x_j}$$ So then would the square be: $$\left[\left(\underline{\nabla}\,\underline{u}\right)^2\right]_i=\frac{\partial u_i}{\partial x_j}\frac{\partial u_i}{\partial x_j}$$ Finally leading to: $$\mathbf{Tr}\left[\left(\underline{\nabla}\,\underline{u}\right)^2\right]_i = \left[\frac{\partial u_i}{\partial x_j}\frac{\partial u_k}{\partial x_j}\right]_{jk}$$ Am I wrong? I'm not very sure on the final line on how a trace will change the suffix notation.