Derivative of the Gram matrix map
= Derivative of the Gram matrix map
{title2=$D(A\mapsto A^tA)_A(H)=H^tA+A^tH$}
For real rectangular <matrices>, expanding $(A+H)^t(A+H)$ leaves the quadratic remainder $H^tH$, whose <matrix norm> is $O(\|H\|^2)$. 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 $A^tA=I$.