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 \(\lambda \in \mathbb{C}\)
Why do you write this with up down indices at all? Wouldn't it be normal to just use down indices, as these are just matrix multiplications we are dealing with, or am I missing something?
Carl (talk) 16:06, 28 July 2015 (CEST)
Quote from Wikipedia: "Einstein notation can be applied in slightly different ways. Typically, each index occurs once in an upper (superscript) and once in a lower (subscript) position in a term; however, the convention can be applied more generally to any repeated indices within a term. When dealing with covariant and contravariant vectors, where the position of an index also indicates the type of vector, the first case usually applies; a covariant vector can only be contracted with a contravariant vector, corresponding to summation of the products of coefficients. On the other hand, when there is a fixed coordinate basis (or when not considering coordinate vectors), one may choose to use only subscripts;"
Oh and: \((E_{ij})_{kl} = \delta_{ik} \delta_{jl} \) or \((E_i{}^j)^k{}_l = \delta_i{}^k \delta^j{}_l \). I'm not sure about this, but lower indices have to stay low, and upper indices have to stay up.
