Re: [線代] 兩個問題

[JohnMash (Paul)]

※ 引述《kuromu (kuromu)》之銘言：
: 看統計的某個定理證明有用到一些性質
: 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

----------------------------------------------------------------------------

另外請問如何證明 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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※ 編輯: JohnMash        來自: 112.104.143.117      (06/02 13:11)
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