Unitary transformation

In mathematics, a unitary transformation is a transformation 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 isomorphism between two Hilbert spaces. In other words, a unitary transformation is a bijective function

where ${\displaystyle H_{1}}$ and ${\displaystyle H_{2}}$ are Hilbert spaces, such that

for all ${\displaystyle x}$ and ${\displaystyle y}$ in ${\displaystyle H_{1}}$.

...
Wikipedia