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# conjugate transpose

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

# 0.0.1 Properties

1. 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

$\displaystyle\det(A^{\ast})$ $\displaystyle=$ $\displaystyle\overline{\det{A}},$ $\displaystyle\operatorname{trace}(A^{\ast})$ $\displaystyle=$ $\displaystyle\overline{\operatorname{trace}{A}},$ $\displaystyle(A^{\ast})^{{-1}}$ $\displaystyle=$ $\displaystyle(A^{{-1}})^{\ast},$ where $\operatorname{trace}$ and $\operatorname{det}$ are the trace and the determinant operators, and ${}^{{-1}}$ is the inverse operator.

4. 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 $A^{\ast}=A^{{\mbox{\scriptsize{H}}}}$).
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.

# See also

## Mathematics Subject Classification

15-00*no label found*15A15

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