# Conjugation of Matrices and Conjugation of Complex Numbers

Are conjugation of matrices and conjugation of complex numbers related?

What I mean is that if $A$ is an $n \times n$ matrix then the conjugation of $A$ by an invertible $n \times n$ matrix $C$ is given by $CAC^{-1}$. On the other hand, if $a + bi$ is a complex number then it's conjugate is $a-bi$. These two operations don't really seem to have anything to do with one another but if they're unrelated why is the same term used to describe the operation?

They are very slightly related in a silly sense. We can embed the complex numbers into the real $2 \times 2$ matrices by sending $a+bi$ to $\left( \begin{smallmatrix} a & b \\ - b & a \end{smallmatrix} \right)$ (note that this is a map of rings). Then complex conjugation corresponds to matrix conjugation by $\left( \begin{smallmatrix} 1 & 0 \\ 0 & -1 \end{smallmatrix} \right)$. That is to say, $$\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix} a & b \\ -b & a \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}^{-1} = \begin{pmatrix} a & -b \\ b & a \end{pmatrix}.$$