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U = U −1 (8.86)
†
jHt
An example of a unitary matrix would be the matrix e , if H was Hermitian.

THEOREM 1
The eigenvalues of a unitary matrix all have magnitude one.
PROOF The eigenvalues and eigenvectors of the unitary matrix satisfy the
usual equations for these quantities; that is:

U v =λ v (8.87)
n n n
Taking the Hermitian conjugate of this equation, we obtain:

v U = v U −1 = v λ (8.88)
†
n n n n
Multiplying Eq. (8.87) on the left by Eq. (8.88), we obtain:

−1
v UU v = vv = λ 2 vv (8.89)
n n n n n n n
from which we deduce the desired result that: λ 2 = 1.
n
A direct corollary of the above theorem is that det( )U = 1 . This can be
proven directly if we remember the result of Pb. 8.15, which states that the
determinant of any diagonalizable matrix is the product of its eigenvalues,
and the above theorem that proved that each of these eigenvalues has unit
magnitude.

THEOREM 2
A transformation represented by a unitary matrix keeps invariant the scalar (dot, or
inner) product of two vectors.
PROOF The matrix U acting on the vectors ϕ and ψ results in two new
vectors, denoted by ϕ' and ψ' and such that:

ϕ ′ = U ϕ (8.90)

ψ ′ = U ψ (8.91)

Taking the Hermitian adjoint of Eq. (8.90), we obtain:

ϕ ′ = ϕ U † = ϕ U −1 (8.92)

© 2001 by CRC Press LLC