This tag is for questions relating to Unitary Matrices which are comprise a class of matrices that have the remarkable properties that as transformations they preserve length, and preserve the angle between vectors.

Definition:A matrix $~U ∈ M_n~$ is said to beunitaryif $~U^∗U = I=UU^∗~$ where $~U^*=\bar U^{\text{T}}~$ (i.e., $~U^∗~$ is complex conjugate transpose of $~U~$)and $~I~$ is the identity matrix.

**Note$~1~$:** If $~U ∈ M_n(\mathbb R)~$ and $~U^{\text{T}}U = I=UU^{\text{T}}~$, then $~U~$ is called real orthogonal.

We can also define Unitary matrix as follows

$U~$ is

unitaryif its conjugate transpose $~U^∗~$ is also its inverse—that is, if $~U^∗=U^{-1}~$.

**Note$~2~$:** Unitary matrices have significant importance in quantum mechanics because they preserve norms, and thus, probability amplitudes.
In this case the Hermitian conjugate of a matrix is denoted by a dagger $~(†)~$ and the equation above becomes $$
{\displaystyle U^{\dagger }U=UU^{\dagger }=I.}$$

- The set of unitary matrices form a group, called the unitary group.

**References:**