# Unitary transformation

unitary mapping

A linear transformation \$A\$ of a unitary space \$L\$ preserving the inner product of vectors, i.e. such that for any vectors \$x\$ and \$y\$ of \$L\$ one has the equality

\$\$(Ax,Ay)=(x,y).\$\$

A unitary transformation preserves, in particular, the length of a vector. Conversely, if a linear transformation of a unitary space preserves the lengths of all vectors, then it is unitary. The eigenvalues of a unitary transformation have modulus 1; the eigenspaces corresponding to different eigenvalues are orthogonal.

A linear transformation \$A\$ of a finite-dimensional unitary space \$L\$ is unitary if and only if it satisfies any of the following conditions:

1) in any orthonormal basis the transformation \$A\$ corresponds to a unitary matrix;

2) \$A\$ maps any orthonormal basis to an orthonormal basis;

3) in \$L\$ there exists an orthonormal basis of eigenvectors of \$A\$, and, moreover, \$A\$ has in this basis a diagonal matrix with diagonal entries of modulus 1.

The unitary transformations of a given unitary space form a group under multiplication of transformations (called the unitary group).