[理工] 證明semipositive definite

[tokyo291 (工口工口)]

X=[X1 X2 ..... Xp] (n*p matrix) n>p

1_n=[1 1 1 1 ... 1]' (n*1 vector)

X1=1_n  X2 ... Xp are all n*1 vectors

H=X[(X'X)^-1]X'

1_n=[1 1 1 1 ... 1]' (n*1 vector)

1.proof that H-(1_n)*(1_n)'/n is semi positive definite

這一題我的證法是用迴歸去想

SSR=(yhat-ybar)'*(yhat-ybar)=y'*(-(1_n)*(1_n)'/n)y>=0

2.proof that diagonal elements of H ,h_i and h_i>=1/n

這一題我也是用迴歸去想h_i=1/n+(x_i-xbar)^2/Sxx

想請問如果要直接用線代的方法證該如何證呢?

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◆ From: 61.227.244.105
※ 編輯: tokyo291        來自: 61.227.244.105       (02/06 00:57)
推 w2i4l2s4o2n4:先說明H-(1_n)*(1_n)'/n是個投影矩陣,那他就是semi-
→ w2i4l2s4o2n4:positive definite