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I am reading a paper "Storage and Solving of Large Sparse Matrix Linear Equations" and I have encounter a terminology and I am asking for meaning of it. A is 6x6 sparse matrix. The paragraph is taken from the paper:

Supposing the large sparse equations as follows:

$Ax = b $

which coefficient matrix is n ranks symmetrical positive definite matrix, supposing

$M = (S^T S) ^{-1 }$

is symmetrical positive definite matrix, which equals to

$A' x'= b', A'= S A S^T , b'=S b, x'= S^{-T} x$ (equation 2)

Using CG method can directly iterate to get the solutions instead of solving (2) firstly.

Here, $x^2$ is x power two. At equation 2 author writes " $A'=S A S ^T$ ". What is this? Could you please explain? Thanks in advance.

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  • $\begingroup$ This is a bit confusing. $S$ is a change-of-coordinates matrix, and you should have $x' = S^{-1}x$. $A'$ is the representation of $A$ in the new coordinates. But the run-on sentence with "which equals to" makes no sense. Ordinarily, one would try to diagonalize $A$ by using $S$, but that doesn't seem to be happening here. You're saying that $A$ is a positive definite symmetric sparse matrix? $\endgroup$ Commented Dec 22, 2019 at 19:46
  • $\begingroup$ $S$ and it's transpose. $\endgroup$
    – user645636
    Commented Dec 22, 2019 at 20:45
  • $\begingroup$ user645636's answer is correct answer. $\endgroup$
    – tahasozgen
    Commented Apr 5, 2020 at 13:18

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