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Difference between revisions of "Adjoint matrix"

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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''
 
''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: $A^* = \overline{(A')}$ where $\bar{\phantom{a}}$ denotes complex conjugation and the $'$ 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: $A^* = \overline{(A')}$ 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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(AB)^* = B^* A^*\,,\ \ \ (A^*)^{-1} = (A^{-1})^*\,,\ \ \ (A^*)^* = 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.
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Adjoint matrices correspond to adjoint linear transformations of [[unitary space]]s with respect to orthonormal bases.
  
 
For references, see [[Matrix]].
 
For references, see [[Matrix]].
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[[Category:Linear and multilinear algebra; matrix theory]]
  
 
{{TEX|done}}
 
{{TEX|done}}

Revision as of 21:12, 17 October 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: $A^* = \overline{(A')}$ 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=33759
This article was adapted from an original article by T.S. Pogolkina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article