Difference between revisions of "Adjoint matrix"
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| − | ''Hermitian adjoint matrix, of a given (rectangular or square) matrix | + | ''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 | + | 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: | 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]]. | |
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| − | + | {{TEX|done}} | |
Revision as of 21:09, 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=12893
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