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Mathematics-Online course: Linear Algebra - Matrices - Matrix Operations

Transpose of and adjoint Matrix


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Interchanging rows and columns of matrix $ A$ we obtain the so called transposed matrix $ B=A^{\operatorname t}$, that is,

$\displaystyle b_{i,j} = a_{j,i}\,
.
$

If we, in addition, replace the complex entries of $ A\in \mathbb{C}^{m\times n}$ by their complex conjugates, then we obtain the so called adjoint matrix $ C=A^\ast = \bar A^{\operatorname t}$, that is,

$\displaystyle c_{i,j} = \bar a_{j,i}\,
.
$

A matrix satisfying $ A=A^{\operatorname t}$ is called symmetric. A matrix satisfying $ A=A^\ast$ is called self-adjoint or Hermitian. For real matrices the notions 'hermitian' and 'symmetric' have the same meaning.

The following rules hold true:


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  automatically generated 4/21/2005