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发帖数: 1037 | 1 Suppose A is an n by n positive definite real matrix. Then there are n
linearly independent vectors v_1,...,v_n such that (v_i,v_j)=A(i,j).
Let v_0 be the projection of v_1 to the space generated by v_2,...v_n.
Then v_0=b_2v_2+...+b_nv_n. To find the coefficients b_2,...b_n, we
solve the equations (v_1-b_2v_2-...-b_nv_n,v_j)=0 for j=2,...,n.
Thus we have (1,-b_2,...,-b_n)A=(C,0,...,0). Let B=A^{-1}. Then we have
(1,-b_2,...,-b_n)=B(C,0,...,0)=C(B(1,1),...,B(n,1)). Thus C=1/B(1,1),
and b_j=-B(j,1 |
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