Unitary transformation

Short description: Endomorphism preserving the inner product

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

$⟨Ux,Uy{⟩}_{H_2}=⟨x,y{⟩}_{H_1}\text{ for all }x,y\in H_1.$

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

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

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 $U$ satisfies $U{U}^{*}=I$ .

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

$⟨Ux,Uy⟩=\overline{⟨x,y⟩}=⟨y,x⟩$

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