I understand that in Ricci calculus $\delta_i^j$ represents the kronecker delta function where $\delta^j_i=1$ if $i=j$ and $0$ otherwise. What I am struggling with though is that I have seen it written for a matrix/(1,1)-tensor $B$;
$$(B')^i_j = B^j_i\delta^{ii}\delta_{jj}$$
Where $B'$ represents the matrix transpose of $B$. I am a bit confused here. What is the interpretation of $\delta^{ii}$? Is there an intuitive way to describe this relationship? Does anyone have any recommendations for learning more about the kronecker delta in Ricci calculus notation in situations like this?
Best Answer
The notation is not standard. In the usual notation, one would write
$$(B')^i{}_j = B^k{}_l \delta^{il}\delta_{kj}$$
Which by Einstein summation convention is just
$$(B')^i{}_j = \sum_k\sum_l B^k{}_l \delta^{il}\delta_{kj}$$.
The symbols $\delta_{ij}$ and $\delta^{ij}$ are defined by $\delta_{ij}=\delta^{ij}=\begin{cases}1&i=j\\0&i\neq j\end{cases}$