# 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?

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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, $$\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix} a & b \\ -b & a \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}^{-1} = \begin{pmatrix} a & -b \\ b & a \end{pmatrix}.$$