D**u
发帖数: 204 | 1 【 以下文字转载自 Science 讨论区 】
发信人: DuGu (火工头陀), 信区: Science
标 题: 矩阵趣题
发信站: BBS 未名空间站 (Fri Sep 7 20:05:00 2007)
Suppose x_1,...,x_n are n positive numbers, prove that the n*n matrix
(1/(x_i+x_j)) (1<=i,j<=n) is positive definite.
Background: this is called Cauchy matrix, and the determinant can be
directed
computed (ref: http://en.wikipedia.org/wiki/Cauchy_determinant ).
What I am asking is: can you still prove the problem by not using
any polynomial-factoring method. Probabilistic/statistical method | i********e
发帖数: 31 | 2 (1) This matrix A = (a_{ij}) = 1/(x_i+x_j) is positive semidefinite
when x_i are positive numbers. A is positive definite only if
x_1, ..., x_n are distinct positive numbers.
(2) Proof: use the following facts
(i) 1/x = \int exp^(-xz)dz for z>=0.
(ii) y'Ay = \sum y_i*y_j*(a_ij)
【在 D**u 的大作中提到】
: 【 以下文字转载自 Science 讨论区 】
: 发信人: DuGu (火工头陀), 信区: Science
: 标 题: 矩阵趣题
: 发信站: BBS 未名空间站 (Fri Sep 7 20:05:00 2007)
: Suppose x_1,...,x_n are n positive numbers, prove that the n*n matrix
: (1/(x_i+x_j)) (1<=i,j<=n) is positive definite.
: Background: this is called Cauchy matrix, and the determinant can be
: directed
: computed (ref: http://en.wikipedia.org/wiki/Cauchy_determinant ).
: What I am asking is: can you still prove the problem by not using
| D**u
发帖数: 204 | 3 Great.
【在 i********e 的大作中提到】
: (1) This matrix A = (a_{ij}) = 1/(x_i+x_j) is positive semidefinite
: when x_i are positive numbers. A is positive definite only if
: x_1, ..., x_n are distinct positive numbers.
: (2) Proof: use the following facts
: (i) 1/x = \int exp^(-xz)dz for z>=0.
: (ii) y'Ay = \sum y_i*y_j*(a_ij)
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