Conjugate transpose: Difference between revisions

Revision as of 15:17, 22 March 2005

Revision as of 15:21, 22 March 2005

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:<math>A=\begin{bmatrix}3+i&2\\

2-2i&i\end{bmatrix}</math>

thenholesky Decomposition

then

:<math>A^*=\begin{bmatrix}3-i&2+2i\\

2&-i\end{bmatrix}.</math>

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* (''AB'')^* = ''B''^*''A''^* for any ''m''-by-''n'' matrix ''A'' and any ''n''-by-''p'' matrix ''B''.

* (''A''^*)^* = ''A'' for any matrix ''A''.

* If ''A'' is a square matrix, then [[determinant|det]] (''A''^*) = (det A)^*, [[trace (matrix)|trace]] (''A''^*) = (trace A)^*, and (''A''^*)^-1 = (''A''^-1)^*.

* <''Ax'',''y''> = <''x'', ''A''^*''y''> for any ''m''-by-''n'' matrix ''A'', any vector ''x'' in '''C'''^''n'' and any vector ''y'' in '''C'''^''m''. Here <.,.> denotes the ordinary Euclidean [[inner product]] (or dot product) on '''C'''^''m'' and '''C'''^''n''.

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Note that there is a general theory of [[adjoint functor]]s in [[category theory]] which includes the previous definition as a special case. See [[John Baez]]' expository article [http://math.ucr.edu/home/baez/week78.html week78] for a discussion of this, and earlier writings for introductory material on category theory.

==External links==

* {{planetmath reference|id=4382|title=Conjugate transpose}}

[[Category:Functional analysis]]

[[Category:Linear algebra]]