# I Pauli matrices and shared eigenvectors

#### Sunny Singh

Summary
For a spin 1/2 particle, why does Sx, Sy and Sz don't share the complete eigenspace even though all of them commute with S^2
We know that S2 commutes with Sz and so they share their eigenspace. Now since S2 also commutes with Sx, as per my understanding, the eigenvectors of S2 and Sz should also be the eigenvectors of Sx. But since the paulic matrices σx and σy are not diagonlized in the eigenbasis of S2, it is clear that S2 and Sx don't share their eigenspace even though they commute with each other. How is that possible? what am i missing?

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#### fresh_42

Mentor
2018 Award
To have common eigenspaces we need that all commute with each other. Every matrix commutes with $I$, but that doesn't mean all matrices have the same eigenspace. And the $S_{xyz}$ do not commute.

#### PeterDonis

Mentor
We know that S2 commutes with Sz and so they share their eigenspace.
Yes.

since S2 also commutes with Sx, as per my understanding, the eigenvectors of S2 and Sz should also be the eigenvectors of Sx.
Your understanding is incorrect. S2 and Sx share an eigenspace, but it's a different eigenspace from the one shared by S2 and Sz. The two eigenspaces must be different because Sx does not commute with Sz.

#### Truecrimson

To say it in another way, commutativity is not transitive. $[A,B] = 0$ and $[B,C]=0$ does not imply that $[A,C]=0$.

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