S𝓍 ≐ ħ͟⎛0 1⎞
     2⎝1 0⎠

Diagonalize the matrix...

First, find the characteristic equation and solve for the eigenvalues.

S𝓍 ≐ ħ͟⎛0 1⎞
     2⎝1 0⎠

ħ͟ ⎛-λ  1⎞
2 ⎝ 1 -λ⎠

    λ² - ħ²/4 = 0 
    λ² = ħ²/4 

    λ = ±ħ/2 

So the eigenvalues are those expected for the measurement of a spin-1/2 component.

The eigenvalue equations are

S𝓍❙Ψ❭ = λ❙Ψ❭

which is represented by

ħ͟⎛0 1⎞⎛a⎞ = λ⎛a⎞
2⎝1 0⎠⎝b⎠    ⎝b⎠