Expansion of the Frobenius norm

Question

this might be very elementary question. I was confused by looking at some different sources when expanding the Frobenius norm into trace. Would these two expressions below always be the same? Or only under certain conditions?
begin{aligned}
left|X-Yright|_{F}^{2}&=operatorname{tr}left(left(X-Yright)left(X-Yright)^{top}right)
end{aligned}
begin{aligned}
left|X-Yright|_{F}^{2}&=operatorname{tr}left(left(X-Yright)^{top}left(X-Yright)right)
end{aligned}
Thanks

José Carlos Santos · Answer

For any two $ntimes n$ matrices $A$ and $B$, $operatorname{tr}(AB)=operatorname{tr}(BA)$ and, in particular,$$operatorname{tr}left(AA^topright)=operatorname{tr}left(A^top Aright).$$

Expansion of the Frobenius norm

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