Proving the Identity to Demonstrating Finite Orthonormal Bases

[franklampard8]

How do I prove that

Ʃ\ket{e_i} \bra{e_i} = I

 

[strangerep]

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.)

 

[franklampard8]

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

 

[strangerep]

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.