Einstein's notation for matrix multiplication $G^{-1} B$

Asked Aug 02 '22 at 17:37

Active Aug 02 '22 at 17:37

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Let us define $G$ as the metric tensor and $B$ as a tensor compatible with metric tensor $G$. How can we denote $G^{-1} \, B$ in Einstein's notation?

I know $g^{ij}$ is every entry of the inverse metric tensor and lower/upper $B_j^i$ is used on the sum of vector/matrices product, but $g^{ij} \, B_j^i$ does not seem right.

I can denote the inverse metric tensor $G^{-1}$ as ⅁ and the desired result would be ⅁${}_{k}^{i} B_{j}^{k}$. However, it is cheating. :-P

asked Aug 02 '22 at 17:37

Bruno Peixoto

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It is difficult to answer this question without more context. Conventionally the answer would be $g^{i j} B_j{}^k$, but Einstein notation is not meant to be used with arbitrary matrices. – Zhen Lin Aug 03 '22 at 03:59
It should be used everywhere. :-P – Bruno Peixoto Aug 03 '22 at 13:34
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Don't get me wrong. I like Einstein notation and even more Penrose graphical notation. But it is meant for the geometry of inner product spaces, not arbitrary matrices. – Zhen Lin Aug 03 '22 at 14:34
Why is your index $k$ ahead $B$ of some indentation space? – Bruno Peixoto Aug 03 '22 at 14:40
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Because otherwise you can't distinguish between $B_j{}^k$ and $B^k{}_j$... – Zhen Lin Aug 03 '22 at 14:45
Oh, I see. I am myself a novice in Einstein's notation. Could you point out some easy-going article? – Bruno Peixoto Aug 03 '22 at 14:50

Einstein's notation for matrix multiplication $G^{-1} B$

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