Use the QR decomposition to prove Hadamard's inequality: if $X=(x_1,cdots,x_n)$, then $$ex |det X|leq prod_{j=1}^n sen{x_j}. eex$$ Equality holds here if and only if the $x_j$ are mutually orthogonal or some $x_j$ are zero.
解答: $$eex ea |det X|^2&=det (X^*X)\ &=det (R^*Q^*QR)\ &=det (R^*R)\ &=prod_{j=1}^n r_{ii}^2\ &leq prod_{j=1}^n sen{x_j}^2, eea eeex$$ where the last inequality follows from the fact that the norm of a vector $geq$ that of is projection (to some subspace).