Show that $(E_{i,j})_{i,j=1}^n$ is a positive matrix.

Question

Let $\{E_{i,j}\}_{i,j=1}^n$ be the matrix units of $M_n(\mathbb{C})$. Consider the matrix $$A=(E_{i,j})_{i,j=1}^n \in M_n(M_n(\mathbb{C})) \cong M_{n^2}(\mathbb{C}).$$

I want to show that this matrix is positive (= self-adjoint and positive eigenvalues).

Attempt:

Let $\xi_1, \dots, \xi_n \in \mathbb{C}^n$. I then calculated $$\left\langle A \begin{pmatrix}\xi_1 \\ \vdots \\ \xi_n\end{pmatrix},\begin{pmatrix}\xi_1 \\ \vdots \\ \xi_n\end{pmatrix} \right\rangle= \sum_{k,l=1}^n (\xi_l)_l\overline{(\xi_k)_k}$$ but I don't see why this expression should be positive. Maybe my calculation is wrong?

Answer 1

Your calculation looks fine. You can show that quantity is non-negative as follows $$\sum_{k,l=1}^n (\xi_l)_l\overline{(\xi_k)_k} = \sum_{l = 1}^{n}\sum_{k = 1}^{n}(\xi_l)_l\overline{(\xi_k)_k} = \left(\sum_{l = 1}^{n}(\xi_l)_l\right)\overline{\left(\sum_{k = 1}^{n}(\xi_k)_k\right)} = \left|\sum_{l = 1}^{n}(\xi_l)_l\right|^2 \ge 0.$$