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An adjoint matrix $A^*$ is equivalent to the matrix conjugate transpose $A$, but I thought conjugate can only be calculated when there's complex entries. How do I go about calculating the conjugate of a matrix with only real entries, $$A=\begin{bmatrix} 1 & 1 \\ 1 & -1 \\ 0 & 1\end{bmatrix},$$ for example.

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    $\begingroup$ The conjugate of a real number is the number itself. $\endgroup$
    – saulspatz
    Apr 12 '18 at 20:04
  • $\begingroup$ But even simpler, the expression for the matrix of the adjoint operator doesn't have conjugation in a real vector space. $\endgroup$
    – user550675
    Apr 12 '18 at 20:13
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The conjugate of a real number is just the number itself. So the adjoint of a real matrix is its transpose. For the example you give, if $$A = \begin{bmatrix}1 & 1 \\ 1 & -1 \\ 0 & 1\end{bmatrix}$$ then $$A^* = \begin{bmatrix}1 & 1 & 0\\ 1 & -1 & 1\end{bmatrix}.$$

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As you know the conjugate of $ a+bi$ is $a-bi$

Thus if $b=0$ the conjugate of $a$ is $a$ itself.

Same goes for real matrices where the conjugate is identical to the matrix itself.

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