Why do Pauli matrices have to be Hermitian and satisfy σ² = I?

[QuasarBoy543298]

TL;DR

questions about Pauli matrices -

why do they need to be Hermitian, what are they trying to measure and why do they need to

satisfy that those matrices squared equals the identity matrices.

Hi :)
I have several questions about the Pauli matrices,
I have seen them when the lecturer showed us Stern-Gerlach experiment
, and we did some really weird assumptions on what we think they should be.

1- why did we assume that all of those matrices should satisfy
σ² = I (the identity matrices)

2- why do they have to be Hermitian?

3- what they are trying to measure? (when we insert <-| for example, we get -<-|, why is that? )
thanks for helping !

 

[haael]

They are a representation of Clifford algebra.

Clifford algebra has operators, abstract objects that have certain properties. Representation is a homomorphism from the operator algebra into matrices. The properties of the matrices reflect the properties of the operators.

https://en.wikipedia.org/wiki/Clifford_algebra#Physics
By using matrices instead of operators you are losing certain general properties, but at the same time you gain certain simplification and also possibility to do numerical approximations.