Basic question about Pauli Rotations

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    Pauli Rotations
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The discussion centers on proving the equation e^{-iAx} = cos(x)I + isin(x)A, given that A^2=I. The key insight is recognizing that the exponential can be expanded using the eigenvalues and eigenvectors of the matrix A. The participant acknowledges a misunderstanding regarding the properties of matrix A, specifically that A^{2n}A = A, which simplifies the derivation process. This clarification resolves the confusion surrounding the proof.

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ArjSiv
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So it's apparently possible to prove that e^{-iAx} = cos(x)I + isin(x)A given that A^2=I.

What I don't understand is how this is supposed to be derived. Any help would be appreciated as this is driving me nuts and this is probably something that is very easy to prove...
 
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Ther are two ways, expand the exponential use the eigenvalues and eigenvectors of the matrix A.
 
Ahh, I knew I was missing something stupid. I didn't realize that A^{2n}A = A.

Thanks :)
 

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