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My course notes are located here http://www.math.ualberta.ca/~mli/courses/current/524/chapter2.pdf

On page 18, in Corollary 2.24, I do not understand how the line $y' = P^{-1}(AP - P'y)$ is obtained.

I was hoping someone might be able to explain how this was done.

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If $x = Py$ then by the product rule ($P'$ denoting the matrix whose entries are the derivatives of the entries of $P$, and similarly for $y'$) one has $$ x' = (Py)' = P' y + Py' $$ and by simple substitution $$ Ax = A(Py), $$ so the statement $x' = Ax$ is equivalent to the statement $P'y + Py' = APy$. If you subtract $P'y$ from both sides of this equation and then left multiply both sides of the resulting equation by $P^{-1}$ you see that $x' = Ax$ is indeed equivalent to the given equation $y' = P^{-1}(AP - P')y$.