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H v =λ v (8.78)
m m m

H v =λ v (8.79)
n n n

and

λ ≠ λ (8.80)
m n

then:

vv = v v = 0 (8.81)
n m m n
PROOF Because the eigenvalues are real, we can write:

v H = v λ (8.82)
n n n

Dot this quantity on the right by the ket v to obtain:
m

v H v = v λ v = λ v v (8.83)
n m n n m n n m

On the other hand, if we dotted Eq. (8.78) on the left with the bra-vector v ,
n
we obtain:

v H v = v λ v = λ v v (8.84)
n m n m m m n m

Now compare Eqs. (8.83) and (8.84). They are equal, or that:

λ vv = λ vv (8.85)
m n m n n m

However, because λ ≠λ , this equality can only be satisfied if vv m = 0,
n
m
n
which is the desired result.

In-Class Exercises
Pb. 8.28 Show that any Hermitian 2 ⊗ 2 matrix has a unique decomposition
into the Pauli spin matrices and the identity matrix.

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