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''of a linear transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010840/a0108401.png" />''
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The linear transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010840/a0108402.png" /> on a Euclidean space (or [[Unitary space|unitary space]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010840/a0108403.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010840/a0108404.png" />, the equality
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<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/a/a010/a010840/a0108405.png" /></td> </tr></table>
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''of a linear transformation  $  A $''
  
between the scalar products holds. This is a special case of the concept of an adjoint linear mapping. The transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010840/a0108406.png" /> is defined uniquely by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010840/a0108407.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010840/a0108408.png" /> is finite-dimensional, then every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010840/a0108409.png" /> has an adjoint <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010840/a01084010.png" />, the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010840/a01084011.png" /> of which in a basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010840/a01084012.png" /> is related to the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010840/a01084013.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010840/a01084014.png" /> in the same basis as follows:
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The linear transformation $  A  ^ {*} $
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on a Euclidean space (or [[Unitary space|unitary space]])  $  L $
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such that for all  $  x, y \in L $,  
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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/a/a010/a010840/a01084015.png" /></td> </tr></table>
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$$
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(Ax, y)  = (x, A  ^ {*} y)
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$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010840/a01084016.png" /> is the matrix adjoint to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010840/a01084017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010840/a01084018.png" /> is the [[Gram matrix|Gram matrix]] of the basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010840/a01084019.png" />.
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between the scalar products holds. This is a special case of the concept of an adjoint linear mapping. The transformation  $  A  ^ {*} $
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is defined uniquely by  $  A $.  
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If  $  L $
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is finite-dimensional, then every  $  A $
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has an adjoint  $  A  ^ {*} $,
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the matrix $  {\mathcal B} $
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of which in a basis  $  e _ {1} \dots e _ {n} $
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is related to the matrix $  {\mathcal A} $
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of $  A $
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in the same basis as follows:
  
In a Euclidean space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010840/a01084020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010840/a01084021.png" /> have the same characteristic polynomial, determinant, trace, and eigen values. In a unitary space, their characteristic polynomials, determinants, traces, and eigen values are [[complex conjugate]]s.
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$$
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{\mathcal B}  = \overline{G}\;  ^ {-1} {\mathcal A}  ^ {*} \overline{G}\; ,
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$$
  
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where  $  {\mathcal A}  ^ {*} $
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is the matrix adjoint to  $  {\mathcal A} $
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and  $  G $
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is the [[Gram matrix|Gram matrix]] of the basis  $  e _ {1} \dots e _ {n} $.
  
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In a Euclidean space,  $  A $
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and  $  A  ^ {*} $
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have the same characteristic polynomial, determinant, trace, and eigen values. In a unitary space, their characteristic polynomials, determinants, traces, and eigen values are [[complex conjugate]]s.
  
 
====Comments====
 
====Comments====
More generally, the phrase  "adjoint transformation"  or  "adjoint linear mappingadjoint linear mapping"  is also used to signify the dual linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010840/a01084022.png" /> of a linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010840/a01084023.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010840/a01084024.png" /> is the space of (continuous) linear functionals on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010840/a01084025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010840/a01084026.png" />. The imbeddings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010840/a01084027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010840/a01084028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010840/a01084029.png" /> connect the two notions. Cf. also [[Adjoint operator|Adjoint operator]].
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More generally, the phrase  "adjoint transformation"  or  "adjoint linear mappingadjoint linear mapping"  is also used to signify the dual linear mapping $  \phi  ^ {*} : M  ^ {*} \rightarrow L  ^ {*} $
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of a linear mapping $  \phi : L \rightarrow M $.  
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Here $  M  ^ {*} $
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is the space of (continuous) linear functionals on $  M $
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and $  \phi  ^ {*} (m  ^ {*} (l)) = m  ^ {*} ( \phi (l)) $.  
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The imbeddings $  L \rightarrow L  ^ {*} $,
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$  M \rightarrow M  ^ {*} $,  
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$  l \mapsto ( \cdot , l) $
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connect the two notions. Cf. also [[Adjoint operator|Adjoint operator]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Reed,  B. Simon,  "Methods of modern mathematical physics" , '''1. Functional analysis''' , Acad. Press  (1972)  pp. Sect. 2</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Reed,  B. Simon,  "Methods of modern mathematical physics" , '''1. Functional analysis''' , Acad. Press  (1972)  pp. Sect. 2</TD></TR></table>

Latest revision as of 16:09, 1 April 2020


of a linear transformation $ A $

The linear transformation $ A ^ {*} $ on a Euclidean space (or unitary space) $ L $ such that for all $ x, y \in L $, the equality

$$ (Ax, y) = (x, A ^ {*} y) $$

between the scalar products holds. This is a special case of the concept of an adjoint linear mapping. The transformation $ A ^ {*} $ is defined uniquely by $ A $. If $ L $ is finite-dimensional, then every $ A $ has an adjoint $ A ^ {*} $, the matrix $ {\mathcal B} $ of which in a basis $ e _ {1} \dots e _ {n} $ is related to the matrix $ {\mathcal A} $ of $ A $ in the same basis as follows:

$$ {\mathcal B} = \overline{G}\; ^ {-1} {\mathcal A} ^ {*} \overline{G}\; , $$

where $ {\mathcal A} ^ {*} $ is the matrix adjoint to $ {\mathcal A} $ and $ G $ is the Gram matrix of the basis $ e _ {1} \dots e _ {n} $.

In a Euclidean space, $ A $ and $ A ^ {*} $ have the same characteristic polynomial, determinant, trace, and eigen values. In a unitary space, their characteristic polynomials, determinants, traces, and eigen values are complex conjugates.

Comments

More generally, the phrase "adjoint transformation" or "adjoint linear mappingadjoint linear mapping" is also used to signify the dual linear mapping $ \phi ^ {*} : M ^ {*} \rightarrow L ^ {*} $ of a linear mapping $ \phi : L \rightarrow M $. Here $ M ^ {*} $ is the space of (continuous) linear functionals on $ M $ and $ \phi ^ {*} (m ^ {*} (l)) = m ^ {*} ( \phi (l)) $. The imbeddings $ L \rightarrow L ^ {*} $, $ M \rightarrow M ^ {*} $, $ l \mapsto ( \cdot , l) $ connect the two notions. Cf. also Adjoint operator.

References

[a1] M. Reed, B. Simon, "Methods of modern mathematical physics" , 1. Functional analysis , Acad. Press (1972) pp. Sect. 2
How to Cite This Entry:
Adjoint linear transformation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Adjoint_linear_transformation&oldid=35199
This article was adapted from an original article by T.S. Pigolkina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article