Talk:Aufgaben:Problem 5

Does anyone like tensor notation and wants to tell me whether this is formally correct?

--Nik (talk) 14:41, 28 July 2015 (CEST)

Ilmanen's solution in Einstein notation

Let \(A \in Z(\mathrm{Mat}_d(\mathbb{C}))\).

Let \(E_{ij}\) be the \(d \times d\)-matrix with \((E_{ij})^k{}_l = \delta^{ij} \delta_{kl} \).

Now, consider $$(E_{ij} A)^k{}_l = (E_{ij})^k{}_m A^m{}_l = \delta^k{}_i \delta^j{}_m A^m{}_l = \delta^k{}_i A^j{}_l$$ but since \(A\) commutes with all compex \(d \times d\)-matrices, this is the same as $$(A E_{ij})^k{}_l = A^k{}_m (E_{ij})^m{}_l = A^k{}_m \delta^m{}_i \delta^j{}_l = \delta^j{}_l A^k{}_i$$

Thus, we have that $$\delta^k{}_i A^j{}_l = \delta^j{}_l A^k{}_i$$ As this holds for any \(1 \leq i,j,k,l \leq d\), we find: $$\forall i \neq j: A^i{}_j = 0 \ \text{and} \ A^i{}_i = A^j{}_j$$ which requires that \(A\) takes the form $$A = \lambda \mathbb{I}_d$$ for some \(\lambda \in \mathbb{C}\)