References: edX online course MIT 8.05.1x Week 4.
Sheldon Axler (2015), Linear Algebra Done Right, 3rd edition, Springer. Chapter 7.
Another important type of operator is the unitary operator , which is defined by the condition that it is surjective and that
for all . That is, a unitary operator preserves the norm of all vectors. The identity matrix is a special case of a unitary operator, as it doesn’t change any vector, but multiplying by any complex number with also preserves the norm, so is another unitary operator.
Because preserves the norm of all vectors, the only vector that can be in the null space of is the zero vector, meaning that is also injective. As it is both injective and surjective, it is invertible.
Theorem 1 For a unitary operator , .
Proof: From its definition and the properties of an adjoint operator, we have
Therefore, so .
Theorem 2 Unitary operators preserve inner products, meaning that for all .
Proof: Since we have
Theorem 3 Acting on an orthonormal basis with a unitary operator produces another orthonormal basis.
Proof: Suppose the orthonormal basis is converted to another set of vectors by :
Thus are an orthonormal set. Since the orthonormal basis spans (by assumption) and the set contains linearly independent orthonormal vectors, is also an orthonormal basis for .
Theorem 4 If one orthonormal basis is converted to another by a unitary operator , then the matrix elements of are the same in both bases.
Proof: This is just a special case of the more general theorem that states that any operator that transforms one set of basis vectors into another has the same matrix elements in both bases. In this case, the proof is especially simple: