unitary (original) (raw)

0.1 Definitions

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  More generally, a unitary transformation is a surjective linear transformation T:U⟶V between two unitary spaces U,V satisfying
  In this entry will restrict to the case of the first , i.e. U=V.
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  In Hilbert spaces unitary transformations correspond precisely to unitary operators.

0.2 Remarks

1. 1. A standard example of a unitary space isℂn with inner product

      ⟨u,v⟩=∑i=1nui⁢vi¯,u,v∈ℂn.	(2)

2. 1. Unitary transformations and unitary matrices are closely related. On the one hand, a unitary matrix defines a unitary transformation of ℂn relative to the inner product (2). On the other hand, the representing matrix of a unitary transformation relative to an orthonormal basis is, in fact, a unitary matrix.
3. 1. A unitary transformation is an automorphism. This follows from the fact that a unitary transformation T preserves the inner-product norm:
      Hence, if
      then by the definition (1) it follows that
      and hence by the inner-product axioms that
      Thus, the kernel of T is trivial, and therefore it is an automorphism.
4. 1. Moreover, relation (3) can be taken as the definition of a unitary transformation. Indeed, using the polarization identity it is possible to show that ifT preserves the norm, then (1) must hold as well.
5. 1. A simple example of a unitary matrix is the change of coordinates matrix between two orthonormal bases. Indeed, letu1,…,un and v1,…,vn be two orthonormal bases, and let A=(Aji) be the corresponding change of basis matrix defined by
      Substituting the above relation into the defining relations for an orthonormal basis,

      ⟨ui,uj⟩	=	δi⁢j,
      ⟨vk,vl⟩	=	δk⁢l,
      we obtain
      ∑i⁢jδi⁢j⁢Aki⁢Alj¯=∑iAki⁢Ali¯=δk⁢l.
      ------------------------------------
      In matrix notation, the above is simply
      as desired.

6. 1. Unitary transformations form a group under composition. Indeed, if S,T are unitary transformations then S⁢T is also surjective and

      ⟨S⁢T⁢u,S⁢T⁢v⟩=⟨T⁢u,T⁢v⟩=⟨u,v⟩
      for every u,v∈V. Hence S⁢T is also a unitary transformation.

7. 1. Unitary spaces, transformations, matrices and operators are of fundamental importance in quantum mechanics.