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''Hermitian adjoint matrix, of a given (rectangular or square) matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010850/a0108501.png" /> over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010850/a0108502.png" /> of complex numbers''
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''Hermitian adjoint matrix, of a given (rectangular or square) matrix $A = \left\Vert{a_{ik}}\right\Vert$ over the field $\mathbb{C}$ of complex numbers''
  
The matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010850/a0108503.png" /> whose entries <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010850/a0108504.png" /> are the complex conjugates of the entries <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010850/a0108505.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010850/a0108506.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010850/a0108507.png" />. Thus, the adjoint matrix coincides with its complex-conjugate transpose: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010850/a0108508.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010850/a0108509.png" /> denotes complex conjugation and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010850/a01085010.png" /> denotes transposition.
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The [[matrix]] $A^*$ whose entries $a^*_{ik}$ are the [[complex conjugate]]s of the entries $a_{ki}$ of $A$, i.e. $a^*_{ik} = \bar a_{ki}$. Thus, the adjoint matrix coincides with its complex-conjugate [[transpose matrix]]: $A^* = \overline{(A')}$ where $\bar{\phantom{a}}$ denotes [[complex conjugation]] and the $'$ denotes transposition.
  
 
Properties of adjoint matrices are:
 
Properties of adjoint matrices are:
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$$
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(A+B)^* = A^* + B^*\,,\ \ \ (\lambda A)^* = \bar\lambda A^*
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$$
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$$
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(AB)^* = B^* A^*\,,\ \ \ (A^*)^{-1} = (A^{-1})^*\,,\ \ \ (A^*)^* = A \ .
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$$
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Adjoint matrices correspond to [[adjoint linear transformation]]s of [[unitary space]]s with respect to [[Orthonormal basis|orthonormal bases]].
  
<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/a010850/a01085011.png" /></td> </tr></table>
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For references, see [[Matrix]].
  
<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/a010850/a01085012.png" /></td> </tr></table>
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[[Category:Linear and multilinear algebra; matrix theory]]
  
Adjoint matrices correspond to adjoint linear transformations of unitary spaces with respect to orthonormal bases.
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{{TEX|done}}
 
 
For references, see [[Matrix|Matrix]].
 

Latest revision as of 16:56, 30 November 2014

Hermitian adjoint matrix, of a given (rectangular or square) matrix $A = \left\Vert{a_{ik}}\right\Vert$ over the field $\mathbb{C}$ of complex numbers

The matrix $A^*$ whose entries $a^*_{ik}$ are the complex conjugates of the entries $a_{ki}$ of $A$, i.e. $a^*_{ik} = \bar a_{ki}$. Thus, the adjoint matrix coincides with its complex-conjugate transpose matrix: $A^* = \overline{(A')}$ where $\bar{\phantom{a}}$ denotes complex conjugation and the $'$ denotes transposition.

Properties of adjoint matrices are: $$ (A+B)^* = A^* + B^*\,,\ \ \ (\lambda A)^* = \bar\lambda A^* $$ $$ (AB)^* = B^* A^*\,,\ \ \ (A^*)^{-1} = (A^{-1})^*\,,\ \ \ (A^*)^* = A \ . $$ Adjoint matrices correspond to adjoint linear transformations of unitary spaces with respect to orthonormal bases.

For references, see Matrix.

How to Cite This Entry:
Adjoint matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Adjoint_matrix&oldid=12893
This article was adapted from an original article by T.S. Pogolkina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article