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. 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 H 1 {displaystyle H_{1}} and H 2 {displaystyle H_{2}} are Hilbert spaces, such that for all x {displaystyle x} and y {displaystyle y} in H 1 {displaystyle H_{1}} . A unitary transformation is an isometry, as one can see by setting x = y {displaystyle x=y} in this formula. In the case when H 1 {displaystyle H_{1}} and H 2 {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 x {displaystyle x} and y {displaystyle y} in H 1 {displaystyle H_{1}} , where the horizontal bar represents the complex conjugate.