Questions tagged [unitary-matrices]

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.

References:

https://en.wikipedia.org/wiki/Unitary_matrix

http://mathworld.wolfram.com/UnitaryMatrix.html