Conjugate transpose: Difference between revisions

Revision as of 23:10, 21 March 2005

Revision as of 15:17, 22 March 2005

Line 1:

In [[mathematics]], the '''conjugate transpose''' or '''adjoint''' of an ''m''-by-''n'' [[matrix (mathematics)|matrix]] ''A'' with [[complex number|complex]] entries is the ''n''-by-''m'' matrix ''A''^* obtained from ''A'' by taking the [[transpose]] and then taking the [[complex conjugate]] of each entry. Formally

:<math>(A^*)[i,j] = \overline{A[j,i]}</math>

for 1 &le; ''i'' &le; ''n'' and 1 &le; ''j'' &le; ''m''. This is a particular case of the [[Hermitian adjoint]] of a [[linear operator]].

This is a particular case of the '''[[Hermitian conjugate]]''' (sometimes called [[Hermitian adjoint]] or just '''adjoint''') [[linear operator]].

More generally, if we have a [[linear map]] ''A'' from a complex vector space ''V'' to another ''W'', the conjugate transpose of ''A'' is the [[complex conjugate linear map|conjugate]] of the [[transpose of a linear map|transpose]] of ''A''. It maps the conjugate dual of ''W'' to the conjugate dual of ''V''.

Alternative names for the conjugate transpose of a matrix are ''adjoint matrix'', ''Hermitian conjugate'', or ''tranjugate''. The conjugate transpose of a matrix ''A'' can be denoted by any of these symbols:

:<math>A^* \qquad A^H \qquad A^\dagger\,. </math>

==Example==

Line 31 ⟶ 33:

* (''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]] (''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''.