[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.1.3

Use the QR decomposition to prove Hadamard's inequality: if $X=(x_1,\cdots,x_n)$, then $$\bex |\det X|\leq \prod_{j=1}^n \sen{x_j}. \eex$$ Equality holds here if and only if the $x_j$ are mutually orthogonal or some $x_j$ are zero.

解答: $$\beex \bea |\det X|^2&=\det (X^*X)\\ &=\det (R^*Q^*QR)\\ &=\det (R^*R)\\ &=\prod_{j=1}^n r_{ii}^2\\ &\leq \prod_{j=1}^n \sen{x_j}^2, \eea \eeex$$ where the last inequality follows from the fact that the norm of a vector $\geq$ that of is projection (to some subspace).