Proving the Identity to Demonstrating Finite Orthonormal Bases

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Discussion Overview

The discussion revolves around proving the identity involving a finite orthonormal basis, specifically the expression Ʃ\ket{ei} \bra{ei} = I, and its implications in linear algebra. The context includes theoretical aspects of linear algebra and the properties of orthonormal bases.

Discussion Character

  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant asks how to prove the identity Ʃ\ket{ei} \bra{ei} = I.
  • Another participant suggests that the proof depends on the context and the definition of the basis vectors e_i, recommending a look at the "spectral theorem" for guidance.
  • A clarification is made that e_i refers to a finite orthonormal basis.
  • It is proposed that expressing an arbitrary vector \psi in terms of the orthonormal basis should lead to the desired result.

Areas of Agreement / Disagreement

Participants generally agree on the importance of context and the definition of the basis vectors, but the discussion does not resolve the proof itself, leaving multiple perspectives on how to approach it.

Contextual Notes

The discussion highlights the dependence on definitions and the need for further exploration of the spectral theorem, but does not resolve specific mathematical steps or assumptions involved in the proof.

franklampard8
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How do I prove that

Ʃ\ket{ei} \bra{ei} = I
 
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franklampard8 said:
How do I prove that
<br /> \def\&lt;{\langle}<br /> \def\&gt;{\rangle}<br /> \sum_i |e_i\&gt;\&lt;e_i| ~=~ 1<br />

It depends on the context, and precisely what your e_i stand for.

As a general suggestion, start by looking up the "spectral theorem" in a linear algebra textbook. (Wikipedia gives an overview, though not the details.)
 
strangerep said:
It depends on the context, and precisely what your e_i stand for.

As a general suggestion, start by looking up the "spectral theorem" in a linear algebra textbook. (Wikipedia gives an overview, though not the details.)

e_i refers to a finite orthonormal basis
 
franklampard8 said:
e_i refers to a finite orthonormal basis

OK, so if you write out how an arbitrary vector \psi is expressed in terms of that orthonormal basis, the desired answer should then follow almost immediately.
 

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