Eigenvalue equations for the S𝓏 operator in a spin-1/2 system:
    S𝓏|+> = +ħ/2|+>
    S𝓏|-> = -ħ/2|->

Matrix form of an operator:
    S𝓏 ≐ ( a b )
         ( c d )

Eigenvalue equations in matrix form:

 ( a b )( 1 ) = +ħ/2 ( 1 )
 ( c d )( 0 )        ( 0 )

 ( a b )( 0 ) = -ħ/2 ( 0 )
 ( c d )( 1 )        ( 1 )

Can show with matrix multiplication operations that
    S𝓏 ≐ ħ/2 ( 1  0 )
             ( 0 -1 )

Some Properties:
     An operator is always diagonal in its own basis.
     Eigenvectors are unit vectirs in their own basis.

    S𝓏 ≐ ħ/2 ( 1  0 )  |+> = ( 1 )   |-> = ( 0 )
             ( 0 -1 )        ( 0 )         ( 1 )


Diagonalizing an Operator
     Diagonize a matrix --> find the eigenvalues and eigenvector
         
     S𝓍 ≐ ħ/2 ( 0 1 )
              ( 1 0 )

     (1) 1⁄2   ⁵⁄ₐ