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Good catch, that's an odd mistake. I would assume the author knows the adjoint isn't equal to the transpose unless the matrix is real.

If you only ever work in R then technically that'd be fine (though notationally awkward). But that's not a good habit to keep since it would introduce pretty insidious bugs if you ever take the adjoint of a complex matrix intending to take the transpose.

But in mathematical derivations, when does a transpose which is not an adjoint ever show up? In many derivations we write transpose knowing that it would be the adjoint if complex numbers are used. That's at least true for most of applied mathematics, statistics, physics, etc.

My point at least was just that it’s a bad idea to use adjoint to construct a column matrix since in that case you likely didn’t intend to take an adjoint, you just wanted a certain shape.

There’s no math or physics equation here with objects transforming under an adjoint representation, it’s just a constructor.

Yes, one may keep that in mind when working something on scratch paper. But when it’s possible for things to be complex, the standard notation is something like A^. In my opinion, it is rather sloppy to write A^T and expect the reader to substitute for the complex case. I can’t remeber where at the moment, but I have definitely seen derivations when the authors explicitly wanted A^T on a complex matrix.

You have to escape your star A^* .

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