# conjugate transpose

Definition If $A$ is a complex matrix, then the conjugate transpose $A^{\ast}$ is the matrix $A^{\ast}=\bar{A}\hskip 1.13811pt^{\mbox{\scriptsize{T}}}\hskip 0.569055pt$, where $\bar{A}$ is the complex conjugate of $A$, and $A\hskip 1.13811pt^{\mbox{\scriptsize{T}}}\hskip 0.569055pt$ is the transpose of $A$.

It is clear that for real matrices, the conjugate transpose coincides with the transpose.

## 0.0.1 Properties

1. 1.

If $A$ and $B$ are complex matrices of same size, and $\alpha,\beta$ are complex constants, then

 $\displaystyle(\alpha A+\beta B)^{\ast}$ $\displaystyle=$ $\displaystyle\overline{\alpha}A^{\ast}+\overline{\beta}B^{\ast},$ $\displaystyle A^{\ast\ast}$ $\displaystyle=$ $\displaystyle A.$
2. 2.

If $A$ and $B$ are complex matrices such that $AB$ is defined, then

 $(AB)^{\ast}=B^{\ast}A^{\ast}.$
3. 3.

If $A$ is a complex square matrix, then

 $\displaystyle\det(A^{\ast})$ $\displaystyle=$ $\displaystyle\overline{\det{A}},$ $\displaystyle\operatorname{trace}(A^{\ast})$ $\displaystyle=$ $\displaystyle\overline{\operatorname{trace}{A}},$ $\displaystyle(A^{\ast})^{-1}$ $\displaystyle=$ $\displaystyle(A^{-1})^{\ast},$

where $\operatorname{trace}$ and $\operatorname{det}$ are the trace and the determinant operators, and ${}^{-1}$ is the inverse operator.

4. 4.

Suppose $\langle\cdot,\cdot\rangle$ is the standard inner product on $\mathbb{C}^{n}$. Then for an arbitrary complex $n\times n$ matrix $A$, and vectors $x,y\in\mathbb{C}^{n}$, we have

 $\langle Ax,y\rangle=\langle x,A^{\ast}y\rangle.$

## Notes

The conjugate transpose of $A$ is also called the adjoint matrix of $A$, the Hermitian conjugate of $A$ (whence one usually writes $A^{\ast}=A\hskip 1.13811pt^{\mbox{\scriptsize{H}}}\hskip 0.569055pt$). The notation $A^{\dagger}$ 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.