## Eigenvalues for X’s Pauli's matrix

Let it be the X coordinate Pauli's matrix:
\begin{array}{ccc}
0 & 1 \\
1 & 0 \end{array}

According to my calculations, it's eigenvectors would require that the spinor components to take the same value, but then, in order to have two orthogonal eigenvectors, we would need the complex components to be orthogonal when doing the dot product.

I choose the eigenvectors ψ_1 =[1, 1] and ψ_2 = [i, i]. Then the dot product must be

ψ_1 · ψ_2 = 1 · i + 1 · i = 0.

That means that orthogonal phases inside the same spinor component must be treated as orthogonal components. Is that true?
 No. Your ψ_2 is proportional your ψ_1; they are not linearly independent. Their dot product of is 2i, not zero. Try [1, 1] and [1, -1] as a complete set of orthogonal eigenvectors.
 Thanks The_duck. With [1, -1] after I pass the Sx operator I'll get [-1, 1], it's the same vector with a diferent phase so it's a valid eigenvector. My mistake was that I forgot the phase factor after the operator. For the [1,1] vector the phase is 0 and for the [-1, 1] it's ∏.