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).

References

[a1]	W.H. Greub, "Linear algebra" , Springer (1975) pp. 338ff