2.3 Unitary Matrices
15
• The inverse of a unitary matrix is obtained by combining complex conjugation
and transposition:
A
−1 = ¯
A
T
(2.17)
Here, T denotes transposition of rows and columns. This result implies that
A
−1
ij = ¯
A ji .
• The inverse and the transpose of a unitary matrix are unitary.
• The product of unitary matrices is a unitary matrix.
• The determinant of a unitary matrix has an absolute value of unity.
To prove the final property, we note that the determinant of a product of matrices is
equal to the product of the determinants of the individual matrices, and we also note
that the determinant does not change upon transposition of a matrix. By definition,
I = A × A −1 , and it then follows:
det
A × A
−1
= det(A) det
A
−1
= det(A) det
¯
A
T
= det(A) det( ¯
A)
= det(A)det(A)
=
det(A)
2 = det(I) = 1
(2.18)
Now consider a function space |f and a linear transformation matrix, A, which
recombines the basis functions to yield a transformed basis set, say |f ′ . Such a
linear transformation of an orthonormal vector space preserves orthonormality if
and only if the transformation matrix A is unitary. Assuming that A is unitary, the
forward implication is easily proven:
f
′
k |f
′
l
=
ij
¯
A ik A jl f i |f j
=
ij
¯
A ik A jl δ ij
=
i
¯
A ik A il
= δ kl
(2.19)
The converse implication is that if a transformation preserves orthonormality, the
corresponding representation matrix will be unitary. Here, the starting point is the
assumption that the basis remains orthonormal after transformation:
i
¯
A ik A il = δ kl
(2.20)
15
• The inverse of a unitary matrix is obtained by combining complex conjugation
and transposition:
A
−1 = ¯
A
T
(2.17)
Here, T denotes transposition of rows and columns. This result implies that
A
−1
ij = ¯
A ji .
• The inverse and the transpose of a unitary matrix are unitary.
• The product of unitary matrices is a unitary matrix.
• The determinant of a unitary matrix has an absolute value of unity.
To prove the final property, we note that the determinant of a product of matrices is
equal to the product of the determinants of the individual matrices, and we also note
that the determinant does not change upon transposition of a matrix. By definition,
I = A × A −1 , and it then follows:
det
A × A
−1
= det(A) det
A
−1
= det(A) det
¯
A
T
= det(A) det( ¯
A)
= det(A)det(A)
=
det(A)
2 = det(I) = 1
(2.18)
Now consider a function space |f and a linear transformation matrix, A, which
recombines the basis functions to yield a transformed basis set, say |f ′ . Such a
linear transformation of an orthonormal vector space preserves orthonormality if
and only if the transformation matrix A is unitary. Assuming that A is unitary, the
forward implication is easily proven:
f
′
k |f
′
l
=
ij
¯
A ik A jl f i |f j
=
ij
¯
A ik A jl δ ij
=
i
¯
A ik A il
= δ kl
(2.19)
The converse implication is that if a transformation preserves orthonormality, the
corresponding representation matrix will be unitary. Here, the starting point is the
assumption that the basis remains orthonormal after transformation:
i
¯
A ik A il = δ kl
(2.20)