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

	[home] [lexicon] [problems] [tests] [courses] [auxiliaries] [notes] [staff]
	Mathematics-Online lexicon:
	Transposed, adjoint, summetric and Hermitian Matrix

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

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

see also:

Keyword: Matrix
Keyword: Adjoint

[Annotations] [Examples]

automatically generated 1/27/2005