Definitions

• A unitary space $V$ is a complex vector space with a distinguished positive definite Hermitian form, $$ \langle -,-\rangle: V\times V \rightarrow \cnums,$$ which serves as the inner product on $V$

• A unitary transformation is a surjective linear transformation $T:V\rightarrow V$ satisfying \begin{equation} \label{eq:def} \langle u,v \rangle = \langle Tu,Tv\rangle,\quad u, v \in V. \end{equation}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
  
  In this entry will restrict to the case of the first definition, 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:
  

  In Hilbert spaces unitary transformations correspond precisely to unitary operators.

Remarks

1. A standard example of a unitary space is $\cnums^n$ with inner product \begin{equation} \label{eq:cprod} \langle u,v \rangle = \sum_{i=1}^n u_i\, \overline{v_i},\quad u,v \in\cnums^n. \end{equation}

2. Unitary transformations and unitary matrices are closely related. On the one hand, a unitary matrix defines a unitary transformation of $\cnums^n$ relative to the inner product (). 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: \begin{equation} \label{eq:def1} \Vert T u \Vert= \Vert u\Vert,\quad u\in V. \end{equation}Hence, if $$Tu=0,$$ then by the definition () it follows that $$\Vert u \Vert = 0,$$ 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 () 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 () must hold as well.
5. A simple example of a unitary matrix is the change of coordinates matrix between two orthonormal bases. Indeed, let $u_1,\ldots, u_n$ 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, \begin{eqnarray*} \langle u_i,u_j\rangle &=& \delta_{ij},\\ \langle v_k,v_l\rangle &=& \delta_{kl}, \end{eqnarray*}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. Unitary transformations form a group under composition. Indeed, if $S, T$ are unitary transformations then $ST$ is also surjective and
   
   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.