# unitary

## 0.1 Definitions

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

 $\langle u,v\rangle=\langle Tu,Tv\rangle,\quad u,v\in V.$ (1)

These are isometries of $V$.

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

 $\langle Tv,Tu\rangle_{V}=\langle v,u\rangle_{U},\quad\;u,v\in U$

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:

 $A^{-1}=\bar{A}^{t}.$
• When $V$ is a Hilbert space, a bounded linear operator $T:V\longrightarrow V$ 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. 1.

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

 $\langle u,v\rangle=\sum_{i=1}^{n}u_{i}\,\overline{v_{i}},\quad u,v\in\mathbb{C% }^{n}.$ (2)
2. 2.

Unitary transformations and unitary matrices are closely related. On the one hand, a unitary matrix defines a unitary transformation of $\mathbb{C}^{n}$ 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. 3.

A unitary transformation is an automorphism. This follows from the fact that a unitary transformation $T$ preserves the inner-product norm:

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

Hence, if

 $Tu=0,$

then by the definition (1) it follows that

 $\|u\|=0,$

and hence by the inner-product axioms that

 $u=0.$

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

4. 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. 5.

A simple example of a unitary matrix is the change of coordinates matrix between two orthonormal bases. Indeed, let $u_{1},\ldots,u_{n}$ and $v_{1},\ldots,v_{n}$ be two orthonormal bases, and let $A=(A^{i}_{j})$ be the corresponding change of basis matrix defined by

 $v_{j}=\sum_{i}A^{i}_{j}\,u_{i},\quad j=1,\ldots,n.$

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

 $\displaystyle\langle u_{i},u_{j}\rangle$ $\displaystyle=$ $\displaystyle\delta_{ij},$ $\displaystyle\langle v_{k},v_{l}\rangle$ $\displaystyle=$ $\displaystyle\delta_{kl},$

we obtain

 $\sum_{ij}\delta_{ij}A^{i}_{k}\overline{A^{j}_{l}}=\sum_{i}A^{i}_{k}\overline{A% ^{i}_{l}}=\delta_{kl}.$

In matrix notation, the above is simply

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

as desired.

6. 6.

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

 $\langle STu,STv\rangle=\langle Tu,Tv\rangle=\langle u,v\rangle$

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

7. 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