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

[Mathematicsresear]

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

 

[PeroK]

Do you mean $\textbf{a.b} = a^{\alpha}b_{\alpha}$?

 

[DrClaude]

See https://en.wikipedia.org/wiki/Einstein_notation#Common_operations_in_this_notation

 

[Mathematicsresear]

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?

 

[PeroK]

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.

 

[BWV]

how would you write the inverse of A^ij is it simply moving the indices downstairs A^ijA_ij=σ ij (where do the indices go, up down or split?)