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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.^[1]

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 ${\textstyle UU^{*}=I}$ ^[2].

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.

References

↑ Hazewinkel, Michiel (1993). Encyclopaedia of Mathematics. Vol. 9. Kluwer Academic Publishers. p. 337. ISBN 978-1-55608-008-1.
↑ Fasi, Massimiliano; Robol, Leonardo (1 July 2021). "Sampling the eigenvalues of random orthogonal and unitary matrices". Linear Algebra and its Applications. 620: 297–321. doi:10.1016/j.laa.2021.02.031.{{cite journal}}: CS1 maint: multiple names: authors list (link)