Is it coincidental that the definition of similar matrices seems to be the conjugate automorphism?
Two matrices $A$ and $B$ are similar if there exists an invertible matrix $P$ for which $B = PAP^{-1}$.
This definition seems presented without any motivation behind it. However, I did notice that it seems to be very much like the automorphism conjugation, $f(x) = axa^{-1}$ for some $a$. Is this just a coincidence, or is there some connection?
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$\begingroup$The motivation is that two matrices are similar if and only if they both represent the same linear transformation with respect to two different bases. So you convert from one basis to another by multiplying by $P^{-1}$, then apply the matrix $A$ which represents the transformation in that other basis, then convert back to the original basis by multiplying by $P$, and get the same result as if you just applied the linear transformation by multiplying by the original matrix $B$, which represents the linear transformation in the original basis.
And that's what conjugation is: transforming something, then doing something with the transformed object, then transforming back again.
So it's not "coincidental" at all.
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