How to Write the Inverse of a Matrix Using Einstein Summation Notation?

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The discussion focuses on writing the dot product using Einstein summation notation, specifically questioning the placement of indices. Participants clarify that the upper index represents a vector and the lower index indicates a covector. The Levi-Civita symbol is mentioned in relation to the cross product, highlighting its significance in tensor operations. There is also a query about expressing the inverse of a matrix using the notation, with confusion about the positioning of indices. Understanding these conventions is crucial for correctly applying Einstein notation in mathematical contexts.
Mathematicsresear
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Homework Statement


I am unsure as to how to write the dot product in terms of the summation notation? May you please explain?

Homework Equations

The Attempt at a Solution

 
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Do you mean ##\textbf{a.b} = a^{\alpha}b_{\alpha}##?
 
PeroK said:
Do you mean ##\textbf{a.b} = a^{\alpha}b_{\alpha}##?
Yes, why is one index is on the top? and the other on the bottom? What about the Levi cevita symbol?
 
Mathematicsresear said:
Yes, why is one index is on the top? and the other on the bottom? What about the Levi cevita symbol?

In addition to the link given in post #3, there must be lots online about the summation convention. Where are you learning this?

The subscript (lower index) indicates the components of a "dual vector" or "covector".

Levi-Civita is used in the cross product.
 
how would you write the inverse of Aij is it simply moving the indices downstairs AijAij=σ ij (where do the indices go, up down or split?)
 

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