Unitary transformation

For other uses, see Transformation (mathematics) (disambiguation).

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.

Formal definition

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

$U:H_{1}\to H_{2}$

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

$\langle Ux,Uy\rangle _{H_{2}}=\langle x,y\rangle _{H_{1}}\quad {\text{ for all }}x,y\in H_{1}.$

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

Unitary operator

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

Relation to unitary matrices

In complex coordinate space $\mathbb {C} ^{n}$ unitary transformations always have the shape

$f\colon \mathbb {C} ^{n}\to \mathbb {C} ^{n},\,x\mapsto U\cdot x$,

where $U\in \mathbb {C} ^{n\times n}$ is a unitary matrix, and the dot before the vector is the matrix-vector-product.

This matrix ${\textstyle U}$ satisfies ${\textstyle UU^{*}=I}$ .

Antiunitary transformation

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

$U:H_{1}\to H_{2}\,$

between two complex Hilbert spaces such that

$\langle Ux,Uy\rangle ={\overline {\langle x,y\rangle }}=\langle y,x\rangle$

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

See also

• Antiunitary
• Orthogonal transformation
• Time reversal
• Unitary group
• Unitary operator
• Unitary matrix
• Wigner's theorem
• Unitary transformations in quantum mechanics