conjugate transpose
Definition If $A$ is a complex matrix, then the conjugate transpose^{} ${A}^{\ast}$ is the matrix ${A}^{\ast}={\overline{A}}^{\text{T}}$, where $\overline{A}$ is the complex conjugate^{} of $A$, and ${A}^{\text{T}}$ is the transpose^{} of $A$.
It is clear that for real matrices, the conjugate transpose coincides with the transpose.
0.0.1 Properties

1.
If $A$ and $B$ are complex matrices of same size, and $\alpha ,\beta $ are complex constants, then
${(\alpha A+\beta B)}^{\ast}$ $=$ $\overline{\alpha}{A}^{\ast}+\overline{\beta}{B}^{\ast},$ ${A}^{\ast \ast}$ $=$ $A.$ 
2.
If $A$ and $B$ are complex matrices such that $AB$ is defined, then
$${(AB)}^{\ast}={B}^{\ast}{A}^{\ast}.$$ 
3.
If $A$ is a complex square matrix^{}, then
$det({A}^{\ast})$ $=$ $\overline{detA},$ $\mathrm{trace}({A}^{\ast})$ $=$ $\overline{\mathrm{trace}A},$ ${({A}^{\ast})}^{1}$ $=$ ${({A}^{1})}^{\ast},$ where $\mathrm{trace}$ and $\mathrm{det}$ are the trace and the determinant^{} operators, and ${}^{1}$ is the inverse operator.

4.
Suppose $\u27e8\cdot ,\cdot \u27e9$ is the standard inner product on ${\u2102}^{n}$. Then for an arbitrary complex $n\times n$ matrix $A$, and vectors $x,y\in {\u2102}^{n}$, we have
$$\u27e8Ax,y\u27e9=\u27e8x,{A}^{\ast}y\u27e9.$$
Notes
The conjugate transpose of $A$ is also called the adjoint matrix of $A$, the Hermitian conjugate of $A$ (whence one usually writes ${A}^{\ast}={A}^{\text{H}}$). The notation ${A}^{\u2020}$ is also used for the conjugate transpose [2]. In [1], ${A}^{\ast}$ is also called the tranjugate of $A$.
References
 1 H. Eves, Elementary Matrix^{} Theory, Dover publications, 1980.
 2 M. C. Pease, Methods of Matrix Algebra, Academic Press, 1965.
See also

•
Wikipedia, http://www.wikipedia.org/wiki/Conjugate_transposeconjugate transpose
Title  conjugate transpose 

Canonical name  ConjugateTranspose 
Date of creation  20130322 13:42:18 
Last modified on  20130322 13:42:18 
Owner  Koro (127) 
Last modified by  Koro (127) 
Numerical id  10 
Author  Koro (127) 
Entry type  Definition 
Classification  msc 1500 
Classification  msc 15A15 
Synonym  adjoint matrix 
Synonym  Hermitian conjugate 
Synonym  tranjugate 
Related topic  Transpose 