View source for Unitary transformation

''unitary mapping''

A [[Linear transformation|linear transformation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095590/u0955901.png" /> of a [[Unitary space|unitary space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095590/u0955902.png" /> preserving the [[Inner product|inner product]] of vectors, i.e. such that for any vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095590/u0955903.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095590/u0955904.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095590/u0955905.png" /> one has the equality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095590/u0955906.png" /></td> </tr></table>

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095590/u0955907.png" /> of a finite-dimensional unitary space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095590/u0955908.png" /> is unitary if and only if it satisfies any of the following conditions:

1) in any orthonormal basis the transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095590/u0955909.png" /> corresponds to a [[Unitary matrix|unitary matrix]];

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095590/u09559010.png" /> maps any orthonormal basis to an orthonormal basis;

3) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095590/u09559011.png" /> there exists an orthonormal basis of eigenvectors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095590/u09559012.png" />, and, moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095590/u09559013.png" /> 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|unitary group]]).



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W.H. Greub,   "Linear algebra" , Springer  (1975)  pp. 338ff</TD></TR></table>