Does the adjoint of an outer product equal to itself?

So (|a><a|)† = |a><a|
 
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Why don't you try to prove it??
 
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A good start might be to try a few test cases. Write down a small (2- or 3-dimensional) complex vector ##|a \rangle## and calculate its outer product with itself. Then check each matrix element to see if the real parts of ##| a \rangle \langle a|## are symmetric and the imaginary parts are anti-symmetric.
 

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