A Understanding of conjugate directions

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Understanding of conjugate directions
I am reading a good paper of J. R. Shewchuk, titled "An introduction to the conjugate gradient method without the agonizing pain", however, I do not fully understand the idea of conjugate directions. As shown in Figure 22a, where the vectors d1 and d2 are not orthogonal. These vectors are transformed by a multiplication with the matrix A and after the transformation we have the corresponding vectors d'1 (=A*d1) and d'2 (=A*d2) as in Figure 22b. If the transformed vectors d'1 and d'2 are orthogonal now, the original vectors d1 and d2 satisfy the condition d2T*A*d1 = 0. The vectors d1 and d2 are then called A-orthogonal or conjuate. So far, so good!

However, I would expect a different condition. The transformed vectors d'1 and d'2 should satisfy the condition d'2T*d'1 = 0. Inserting there d'1 = A*d1 and d'2 = A*d2 would yield the condition d2T*AT*A*d1=0. This condition is different from the condition for the A-orthogonality. And I don't understand why ... ?
 
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Are you sure it should be the same matrix? Or just any matrix, like ##d_2^\tau B d_1## with ##B=A^\tau A##. Another way out is to search a matrix ##A## such that ##d_1'=d_1A## and ##d_2'=Ad_2## are orthogonal.
 
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