I Pauli matrices

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 !!!
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.

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.

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