conjugate transpose

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

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

Properties

1. If $A$ and $B$ are complex matrices of same size, and $\alpha,\beta$ are complex constants, then \begin{eqnarray*} (\alpha A + \beta B)^\ast &=& \overline{\alpha} A^\ast + \overline{\beta} B^\ast,\\ A^{\ast\ast} &=& A. \end{eqnarray*}
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 \begin{eqnarray*} \det (A^\ast) &=& \overline{ \det{A}}, \\ \operatorname{trace}(A^\ast) &=& \overline{ \operatorname{trace}{A}}, \\ (A^\ast)^{-1} &=& (A^{-1})^\ast, \end{eqnarray*}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 $\sC^n$ . Then for an arbitrary complex $n\times n$ matrix $A$ , and vectors $x,y\in \sC^n$ , 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 ). 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

• Wikipedia, conjugate transpose