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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$\begingroup$They are not related at all. Actually, you are comparing "conjugation" with "conjugate". The word "conjugate" is also applied to matrices as a synonym of "adjoint", and in that case it is a direct generalization of the complex conjugate.
Unfortunately, math is full of incoherent terminology.
$\endgroup$ 1 $\begingroup$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, $$\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix} a & b \\ -b & a \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}^{-1} = \begin{pmatrix} a & -b \\ b & a \end{pmatrix}.$$
$\endgroup$ $\begingroup$I know this is super old but there's a good answer that nobody gave here. Both operations are automorphisms. That is, the sum of two conjugates is the conjugate of the sum, and the conjugate of the product is the product of the conjugates. This holds in both cases.
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