unitary
0.1 Definitions

•
A unitary space $V$ is a complex vector space with a distinguished positive definite^{} Hermitian form^{},
$$\u27e8,\u27e9:V\times V\to \u2102,$$ which serves as the inner product^{} on $V$.

•
A unitary transformation is a surjective^{} linear transformation $T:V\to V$ satisfying
$$\u27e8u,v\u27e9=\u27e8Tu,Tv\u27e9,u,v\in V.$$ (1) These are isometries^{} of $V$.

•
More generally, a unitary transformation is a surjective linear transformation $T:U\u27f6V$ between two unitary spaces $U,V$ satisfying
$${\u27e8Tv,Tu\u27e9}_{V}={\u27e8v,u\u27e9}_{U},u,v\in U$$ In this entry will restrict to the case of the first , i.e. $U=V$.

•
A unitary matrix is a square complexvalued 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 $T:V\u27f6V$ 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 ${\u2102}^{n}$ with inner product
$$\u27e8u,v\u27e9=\sum _{i=1}^{n}{u}_{i}\overline{{v}_{i}},u,v\in {\u2102}^{n}.$$ (2) 
2.
Unitary transformations and unitary matrices are closely related. On the one hand, a unitary matrix defines a unitary transformation of ${\u2102}^{n}$ 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 innerproduct 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 innerproduct 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,
$\u27e8{u}_{i},{u}_{j}\u27e9$ $=$ ${\delta}_{ij},$ $\u27e8{v}_{k},{v}_{l}\u27e9$ $=$ ${\delta}_{kl},$ 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
$$\u27e8STu,STv\u27e9=\u27e8Tu,Tv\u27e9=\u27e8u,v\u27e9$$ 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  20130322 12:02:01 
Last modified on  20130322 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 1500 
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 