unitary

0.1 Definitions

• A unitary space $V$ is a complex vector space with a distinguished positive definite Hermitian form,

  $$⟨-,-⟩:V\times V\to ℂ,$$

  which serves as the inner product on $V$.

• A unitary transformation is a surjective linear transformation $T:V\to V$ satisfying

  $$⟨u,v⟩=⟨Tu,Tv⟩,u,v\in V.$$ (1)

  These are isometries of $V$.

• More generally, a unitary transformation is a surjective linear transformation $T:U⟶V$ between two unitary spaces $U,V$ satisfying

  $${⟨Tv,Tu⟩}_V={⟨v,u⟩}_U,u,v\in U$$

  In this entry will restrict to the case of the first , i.e. $U=V$.

• A unitary matrix is a square complex-valued matrix, $A$, whose inverse is equal to its conjugate transpose:

  $$A^{-1}={\overline{A}}^t.$$

• When $V$ is a Hilbert space, a bounded linear operator $T:V⟶V$ is said to be a unitary operator if its inverse is equal to its adjoint:

  $$T^{-1}=T^{*}$$

  In Hilbert spaces unitary transformations correspond precisely to unitary operators.

0.2 Remarks

1. A standard example of a unitary space is ${ℂ}^n$ with inner product

   $$⟨u,v⟩=\sum _{i=1}^n{u}_i\overline{v_i},u,v\in {ℂ}^n.$$ (2)

2. Unitary transformations and unitary matrices are closely related. On the one hand, a unitary matrix defines a unitary transformation of ${ℂ}^n$ 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. A unitary transformation is an automorphism. This follows from the fact that a unitary transformation $T$ preserves the inner-product norm:

   $$\parallel Tu\parallel =\parallel u\parallel ,u\in V.$$ (3)

   Hence, if

   $$Tu=0,$$

   then by the definition (1) it follows that

   $$\parallel u\parallel =0,$$

   and hence by the inner-product axioms that

   $$u=0.$$

   Thus, the kernel of $T$ is trivial, and therefore it is an automorphism.

4. Moreover, relation (3) can be taken as the definition of a unitary transformation. Indeed, using the polarization identity it is possible to show that if $T$ preserves the norm, then (1) must hold as well.

5. A simple example of a unitary matrix is the change of coordinates matrix between two orthonormal bases. Indeed, let $u_1,\mathrm{\dots },u_n$ and $v_1,\mathrm{\dots },v_n$ be two orthonormal bases, and let $A=(A_j^i)$ be the corresponding change of basis matrix defined by

   $$v_j=\sum _i{A}_j^i{u}_i,j=1,\mathrm{\dots },n.$$

   Substituting the above relation into the defining relations for an orthonormal basis,

   $⟨u_i,u_j⟩$ $=$ ${\delta }_{ij},$

   $⟨v_k,v_l⟩$ $=$ ${\delta }_{kl},$

   we obtain

   $$\sum _{ij}{\delta }_{ij}{A}_k^i\overline{A_l^j}=\sum _i{A}_k^i\overline{A_l^i}={\delta }_{kl}.$$

   In matrix notation, the above is simply

   $$A{\overline{A}}^t=I,$$

   as desired.

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

   $$⟨STu,STv⟩=⟨Tu,Tv⟩=⟨u,v⟩$$

   for every $u,v\in V$. Hence $ST$ is also a unitary transformation.

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

Title	unitary
Canonical name	Unitary
Date of creation	2013-03-22 12:02:01
Last modified on	2013-03-22 12:02:01
Owner	asteroid (17536)
Last modified by	asteroid (17536)
Numerical id	21
Author	asteroid (17536)
Entry type	Definition
Classification	msc 47D03
Classification	msc 47B99
Classification	msc 47A05
Classification	msc 46C05
Classification	msc 15-00
Synonym	complex inner product space
Related topic	EuclideanVectorSpace2
Related topic	PauliMatrices
Defines	unitary space
Defines	unitary matrix
Defines	unitary transformation
Defines	unitary operator
Defines	unitary group