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

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Discussion Overview

The discussion revolves around writing the inverse of a matrix using Einstein summation notation, with a focus on the representation of dot products and the roles of indices in this notation. Participants explore the implications of index placement and the use of the Levi-Civita symbol.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Conceptual clarification

Main Points Raised

  • One participant seeks clarification on how to express the dot product in summation notation.
  • Another participant proposes that the dot product can be expressed as ##\textbf{a.b} = a^{\alpha}b_{\alpha}##.
  • There is a question regarding the placement of indices, specifically why one index is on the top and the other on the bottom.
  • Discussion includes the relevance of the Levi-Civita symbol, particularly in relation to the cross product.
  • A participant inquires about how to write the inverse of a matrix, questioning the placement of indices in the expression for the inverse.

Areas of Agreement / Disagreement

Participants express uncertainty regarding the conventions of index placement and the implications for the inverse of a matrix, indicating that multiple views and questions remain unresolved.

Contextual Notes

There are limitations in the discussion regarding the assumptions about the notation and the definitions of dual vectors or covectors, which are not fully explored.

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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