Difference between revisions of "Unitary transformation"
From Encyclopedia of Mathematics
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''unitary mapping'' | ''unitary mapping'' | ||
| − | A [[Linear transformation|linear transformation]] | + | A [[Linear transformation|linear transformation]] $A$ of a [[Unitary space|unitary space]] $L$ preserving the [[Inner product|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 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 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 | + | 1) in any orthonormal basis the transformation $A$ corresponds to a [[Unitary matrix|unitary matrix]]; |
| − | 2) | + | 2) $A$ maps any orthonormal basis to an orthonormal basis; |
| − | 3) in | + | 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|unitary group]]). | The unitary transformations of a given unitary space form a group under multiplication of transformations (called the [[Unitary group|unitary group]]). | ||
Latest revision as of 11:38, 1 August 2014
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).
Comments
References
| [a1] | W.H. Greub, "Linear algebra" , Springer (1975) pp. 338ff |
How to Cite This Entry:
Unitary transformation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Unitary_transformation&oldid=32651
Unitary transformation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Unitary_transformation&oldid=32651
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article