Abstract
It is shown that every solution to the equation \( X\bar X = \bar XX \) can be reduced by a real orthogonal similarity transformation to a block triangular form with diagonal blocks of orders one and two. If the solution X is a normal matrix, then its block triangular form is actually a block diagonal. In this case, the form of the diagonal blocks is found, yielding new proof of the recent results of Goodson and Horn.
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