Eigenvalue Problem for 2×2 Hermitian Matrices

a

2

b

-1

c

1

d

-1



a	c+d
c-d	b



α

β

 = λ 

α

β





2.00	1.00-1.00
1.00+1.00	-1.00



0.86-0.36

0.34+0.14

 = 2.56 

0.86-0.36

0.34+0.14





2.00	1.00-1.00
1.00+1.00	-1.00



-0.34-0.14

0.86-0.36

 = -1.56 

-0.34-0.14

0.86-0.36



An

n×n

Hermitian matrix (

M

ji

=

*

M

ji

) has

n

real eigenvalues and

n

mutually orthogonal eigenvectors, which can be chosen to be normalized. This Demonstration considers the case of

2×2

Hermitian matrices, which has important applications in the study of two-level quantum systems. For a selected

2×2

Hermitian matrix, the graphic shows the equations satisfied by the two eigenvalues, with their corresponding orthonormalized eigenvectors.