# Denotation of A'

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.

• 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? Commented Dec 22, 2019 at 19:46
• $S$ and it's transpose.
– user645636
Commented Dec 22, 2019 at 20:45
• user645636's answer is correct answer. Commented Apr 5, 2020 at 13:18