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 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. | + | 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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(AB)^* = B^* A^*\,,\ \ \ (A^*)^{-1} = (A^{-1})^*\,,\ \ \ (A^*)^* = A \ . | (AB)^* = B^* A^*\,,\ \ \ (A^*)^{-1} = (A^{-1})^*\,,\ \ \ (A^*)^* = A \ . | ||
$$ | $$ | ||
| − | Adjoint matrices correspond to adjoint linear | + | Adjoint matrices correspond to [[adjoint linear transformation]]s of [[unitary space]]s with respect to [[Orthonormal basis|orthonormal bases]]. |
For references, see [[Matrix]]. | For references, see [[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.
Adjoint matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Adjoint_matrix&oldid=35178