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

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

   $$⟨Ax,y⟩=⟨x,A^{\ast }y⟩.$$

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 $A^{†}$ is also used for the conjugate transpose [2]. In [1], $A^{\ast }$ is also called the tranjugate of $A$.

See also

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

Title	conjugate transpose
Canonical name	ConjugateTranspose
Date of creation	2013-03-22 13:42:18
Last modified on	2013-03-22 13:42:18
Owner	Koro (127)
Last modified by	Koro (127)
Numerical id	10
Author	Koro (127)
Entry type	Definition
Classification	msc 15-00
Classification	msc 15A15
Synonym	adjoint matrix
Synonym	Hermitian conjugate
Synonym	tranjugate
Related topic	Transpose