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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 and are Hilbert spaces, such that

for all and in .

A unitary transformation is an isometry, as one can see by setting in this formula.

In the case when and 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 and in , where the horizontal bar represents the complex conjugate.