If I have a matrix representation $\phi$: $G → GL(d, \mathbb{C})$ and make a direct sum with its complex conjugate, how do I prove that this sum is matrix representation of G over $\mathbb{R}$? I would perhaps prove equivalence with some real matrix representation $\psi$: $\phi$: $G → GL(2d, \mathbb{R})$, but just don´t know the steps.
2026-03-31 17:33:39.1774978419
Show that direct sum of a representation with its complex conjugate is defined over $\mathbb{R}$
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Hint: $$ \frac 12 \pmatrix{I & iI\\-iI & -I} \pmatrix{A + iB & 0\\0 & A - iB} \pmatrix{I & iI\\-iI & -I} = \pmatrix{A & -B\\B & A}. $$