Derivative of the Gram matrix map ($D(A\mapsto A^{t}A{)}_A(H)=H^{t}A+A^{t}H$)

For real rectangular matrices, expanding $(A+H{)}^t(A+H)$ leaves the quadratic remainder $H^{t}H$, whose matrix norm is $O(\Vert H{\Vert }^2)$. Thus the displayed Fréchet derivative is a linear map in the increment $H$. At an orthogonal matrix, its kernel consists of increments $H=A\Omega$ with $\Omega$ skew-symmetric. This is the tangent-space condition obtained by differentiating $A^{t}A=I$.