# Conjugate Transpose

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A Conjugate Transpose is a [[]] that ...

**See:**Moore–Penrose Pseudoinverse, Mathematics, Matrix (Mathematics), Complex Number, Transpose, Complex Conjugate, Linear Algebra, Dagger (Typography), Quantum Mechanics.

## References

### 2015

- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/conjugate_transpose Retrieved:2015-3-1.
- In mathematics, the
**conjugate transpose**or '*Hermitian transpose of an*m*-by-*n matrix*A*with complex entries is the*n*-by-m*matrix*AA^{*}obtained from*by taking the transpose and then taking the complex conjugate of each entry (i.e., negating their imaginary parts but not their real parts). The conjugate transpose is formally defined by : [math]\displaystyle{ (\boldsymbol{A}^*)_{ij} = \overline{\boldsymbol{A}_{ji}} }[/math] where the subscripts denote the*i*,*j*-th entry, for 1 ≤*i*≤*n and 1 ≤*j*≤*m*, and the overbar denotes a scalar complex conjugate. (The complex conjugate of [math]\displaystyle{ a + bi }[/math] , where*a*and*b*are reals, is [math]\displaystyle{ a - bi }[/math] .)This definition can also be written as : [math]\displaystyle{ \boldsymbol{A}^* = (\overline{\boldsymbol{A}})^\mathrm{T} = \overline{\boldsymbol{A}^\mathrm{T}} }[/math] where [math]\displaystyle{ \boldsymbol{A}^\mathrm{T} \,\! }[/math] denotes the transpose and [math]\displaystyle{ \overline{\boldsymbol{A}} \,\! }[/math] denotes the matrix with complex conjugated entries.

Other names for the conjugate transpose of a matrix are

**Hermitian conjugate**, bedaggered matrix,**adjoint matrix**or transjugate. The conjugate transpose of a matrix*A*can be denoted by any of these symbols:- [math]\displaystyle{ \boldsymbol{A}^* \,\! }[/math] or [math]\displaystyle{ \boldsymbol{A}^\mathrm{H} \,\! }[/math] , commonly used in linear algebra.
- [math]\displaystyle{ \boldsymbol{A}^\dagger \,\! }[/math] (sometimes pronounced as "
*A*dagger"), universally used in quantum mechanics. - [math]\displaystyle{ \boldsymbol{A}^+ \,\! }[/math] , although this symbol is more commonly used for the Moore–Penrose pseudoinverse.

- In some contexts, [math]\displaystyle{ \boldsymbol{A}^* \,\! }[/math] denotes the matrix with complex conjugated entries, and the conjugate transpose is then denoted by [math]\displaystyle{ \boldsymbol{A}^{*\mathrm{T}} \,\! }[/math] or [math]\displaystyle{ \boldsymbol{A}^{\mathrm{T}*} \,\! }[/math] .

- In mathematics, the