Understanding of conjugate directions

[Ironphys]

TL;DR Summary

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 d₂^T*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'₂^T*d'₁ = 0. Inserting there d'₁ = A*d₁ and d'₂ = A*d₂ would yield the condition d₂^T*A^T*A*d₁=0. This condition is different from the condition for the A-orthogonality. And I don't understand why ... ?

 

[fresh_42]

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.