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 be unitary if $~U^∗U = I=UU^∗~$ 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 unitary if 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.
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