线性代数拔尖线性代数随机推荐教材习题蒲和平非数习题解答综合题 9*线性代数综合分类讨论不看本题本题讨论已掌握查看解析收入错题#16034设 f(x1,x2,⋯ ,xn)=∑i,j=1naijxixjf\left( {{x}_{1},{x}_{2},\cdots ,{x}_{n}}\right) = \mathop{\sum }\limits_{i,j = 1}^{n}{a}_{ij}{x}_{i}{x}_{j}f(x1,x2,⋯,xn)=i,j=1∑naijxixj ,求积分 ∫0+∞f(x1e−t,x2e−2t,⋯ ,xne−nt)dt{\int }_{0}^{+\infty }f\left( {{x}_{1}{e}^{-t},{x}_{2}{e}^{-{2t}},\cdots ,{x}_{n}{e}^{-{nt}}}\right) \mathrm{d}t∫0+∞f(x1e−t,x2e−2t,⋯,xne−nt)dt ,并证明当 aij=1i+j{a}_{ij} = \frac{1}{i + j}aij=i+j1 时,实对称矩阵 A=(aij)n×n\mathbf{A} = {\left( {a}_{ij}\right) }_{n \times n}A=(aij)n×n 与 B=(aiji+j)n×n\mathbf{B} = {\left( \frac{{a}_{ij}}{i + j}\right) }_{n \times n}B=(i+jaij)n×n 都是正定矩阵.