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The Reverse-order law for the Moore--Penrose inverse is false in general. TakeThenA sufficient condition isso that $A$ has full column rank and $B$ has full row rank. Indeed $A^{†}A=I_n$ and $B{B}^{†}=I_n$; these identities make $B^{†}{A}^{†}$ satisfy all four Penrose equations for $AB$.

The Hessian of $-h$ is $M=A{Q}^{-1}{A}^T$. Thus $-h$ is strongly convex exactly when $A$ has full row rank. In that case one may useIn all cases, the gradient has Lipschitz constant bounded by

The identity $(AB{)}^{†}=B^{†}{A}^{†}$ does not hold for arbitrary matrices. A sufficient condition is that $A$ have full column rank and $B$ have full row rank.