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

such that

It is a linear isometry

Unitary operator

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 unitary transformations always have the shape

1. ,

is a unitary matrix, and the dot before the vector is the matrix-vector-product.

.

Antiunitary transformation

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

between two complex Hilbert spaces such that

, 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

References