Conjugate transpose: Difference between revisions

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

for 1 &le; ''i'' &le; ''n'' and 1 &le; ''j'' &le; ''m''.

This definition can also be written as

:<math> A^*H \equiv {\overline A}^{T}</math>

where <math>A^T \,\!</math> denotes the transpose and <math> \overline A \,\!</math> denotes the matrix with complex conjugated entries.

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^* \,\!</math>

* <math>A^H \,\!</math>, commonly used in [[linear algebra]]

* <math>A^\dagger \,\!</math>, universally used in [[quantum mechanics]]

In some contexts* <math>A^* \,\!</math> (which can be used toalso denote the complex conjugate, so care must be taken not to confuse notations.however)

In some contexts <math>A^* \,\!</math> can be used to denote the complex conjugate so care must be taken not to confuse notations.

==Example==

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

then

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

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