作者JohnMash (Paul)
看板Math
标题Re: [线代] 两个问题
时间Thu Jun 2 08:25:42 2011
※ 引述《kuromu (kuromu)》之铭言：
: 看统计的某个定理证明有用到一些性质
: if A real symmetric matrix is positive semidefinite,and if an element on the
: principal diagonal is zero,then all elements in the row and all elements in
: the column are zero.
A=A^t
x^t A x ≧0 for all x
then A_{kk}≧0 for all k
Suppose A_{ii}=0 for some i
--------------------------------------------------------------------------
Suppose A_{ij}≧0
Take x_i=r, x_j=-1/r, and x_k=0 for k≠i,j
then x^t A x = x_m A_{mn} x_n = A_{jj} /r^2 - A_{ij} - A_{ji}≧0
for all r
Hence, 2A_{ij}≦A_{jj}/r^2 for all r
A_{ij}=0
----------------------------------------------------------------------------
Suppose A_{ij}≦0
Take x_i=r, x_j=1/r, and x_k=0 for k≠i,j
then x^t A x = x_m A_{mn} x_n = A_{jj} /r^2 + A_{ij} + A_{ji}≧0
for all r
Hence, -2A_{ij}≦A_{jj}/r^2 for all r
A_{ij}=0
----------------------------------------------------------------------------
另外请问如何证明 rank(A1+A2+...)≦rank(A1)+rank(A2)+...
If x in the range(A1+A2)
then
(A1+A2).y = x = A1.y+A2.y
that is, x in the range(A1)+range(A2)
range(A1+A2) is contained in range(A1)+range(A2)
However, dim(V+W)≦dim(V)+dim(W)
rank(A1+A2)≦dim(range(A1)+range(A2))≦rank(A1)+rank(A2)
--
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◆ From: 112.104.97.155
※ 编辑: JohnMash 来自: 112.104.143.117 (06/02 13:11)
1F：→ kuromu :谢谢 06/02 21:31