Unitary transformation

In mathematics, a unitary transformation is a linear isomorphism that preserves the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation.

More precisely, a unitary transformation is an isometric isomorphism between two inner product spaces (such as Hilbert spaces). In other words, a unitary transformation is a bijective function

between two inner product spaces, ${\displaystyle H_{1}}$ and ${\displaystyle H_{2},}$ such that

It is a linear isometry, as one can see by setting ${\displaystyle x=y.}$

In the case when ${\displaystyle H_{1}}$ and ${\displaystyle H_{2}}$ are the same space, a unitary transformation is an automorphism of that Hilbert space, and then it is also called a unitary operator.

A closely related notion is that of antiunitary transformation, which is a bijective function

between two complex Hilbert spaces such that

for all ${\displaystyle x}$ and ${\displaystyle y}$ in ${\displaystyle H_{1}}$, where the horizontal bar represents the complex conjugate.