How are eigenvectors and eigenvalues expressed in the Bloch sphere?

Question

I'm relatively new to the subject of quantum computing, and I recently came across the idea of eigenvalues and eigenvectors. I believe I understand the relationship between the two, where eigenvalues determine the factor by which an eigenvector is stretched, and eigenvectors represent the direction that the transformation is pointing towards, but I was wondering how exactly eigenvectors and eigenvalues would be expressed in the Bloch sphere.

DaftWullie · Accepted Answer

The eigenvectors for a one-qubit unitary are two orthogonal vectors. As such, on the Bloch sphere, they are visualised as a single axis (going through the origin). (Remember that angles on the Bloch sphere are doubled so orthogonal states are an angle $pi$ different on the Bloch sphere, i.e. opposite directions along the same axis.)
The eigenvalue (or, more precisely, the relative angle between the two eigenvalues) is the angle of rotation around that axis.

glS · Answer

A state $rho$ with Bloch sphere coordinates $newcommand{bs}[1]{boldsymbol{#1}}bs requiv (x,y,z)$ has the form
$$rho = frac{I + bs rcdotbs sigma}{2}equiv frac{I+xsigma_x + y sigma_y + zsigma_z}{2}, $$
with $sigma_x,sigma_y,sigma_z$ the Pauli matrices.
Computing the eigenvalues (eigenvectors) of $rho$ thus amounts to computing those of $bs rcdotbssigma$. Observe that
$$bs rcdotbs sigma=begin{pmatrix}z & x-iy  x+iy & -z,end{pmatrix}$$
and thus the eigenvalues are $lambda_pm = pmsqrt{-det(bs rcdotbs sigma)}=pm|bs r|$. The corresponding eigenvectors are then seen to be
$$lvertlambda_pmrangle = frac{1}{sqrt{2|bs r|(|bs r|mp z)}}begin{pmatrix}x-iy  pm |bs r| - zend{pmatrix}.$$
The vectors in the Bloch sphere corresponding to $lvertlambda_pmrangle$ have coordinates
$$begin{cases}
x_pm &=& pm x/ |bs r|, 
y_pm &=& pm y/ |bs r|, 
z_pm &=& pm z/ |bs r|.
end{cases}$$
In other words, the eigenvectors of $bs rcdotbssigma$ correspond to the two unit vectors in the Bloch sphere along the same direction as $rho$.
The eigenvectors of $rho$ are then clearly the same as those of $bs rcdotbs sigma$, while its eigenvalues are $(1pmlambda_pm)/2$.

How are eigenvectors and eigenvalues expressed in the Bloch sphere?

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