Mathematics-Online lexicon: Transposed, adjoint, summetric and Hermitian Matrix

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	Mathematics-Online lexicon:
	Transposed, adjoint, summetric and Hermitian Matrix

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overview

Interchanging rows and columns of matrix

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:

$(A B)^{\operatorname t}= B^{\operatorname t}A^{\operatorname t}$ and $(A B)^\ast = B^\ast A^\ast$ ,
$(A^{\operatorname t})^{-1} = (A^{-1})^{\operatorname t}$ and $(A^\ast)^{-1} = (A^{-1})^\ast$ and

Annotation:

Proof of the Rules

[Examples] [Links]

automatically generated 1/27/2005