Explanation of matrix similarity $B=P^{-1}AP$ So two matrices $A,B$ are similar iff there exists some matrix $P$ such that $B=P^{-1}AP$ but could someone explain what this really means? I understand that P is the change of basis matrix but could someone maybe try to simply explain what this multiplication is doing to A?
 A: Algebraically the construction with $PXP^{-1}$ is highly useful and called the conjugation of $X$ by $P$ and it shows up in group theory as well. This is because conjugation is an automorphism so it preserves many of the interesting properties of the space you're working with. One way to think about automorphism of a vector space is as a change of basis since it preserves the space but changes the coordinates you use to represent the vectors in it. Automorphisms can also be thought of as a relabeling of the elements in a space, so they're the same thing just with new names.
Another useful property of conjugates is that if $B=PXP^{-1}$ then $B^n = PX^nP^{-1}$. This will be useful later as you can sometimes choose a $P$ that makes $B$ a diagonal matrix, thus simplifying the calculation of $B^n$ which allows you to then calculate $X^n$ using significantly less operations than doing the full matrix multiplication so it has a practical side as well.
A: The map $B \mapsto PBP^{-1}$ is called the conjugation map. Is essentially an automorfism (biyective operator), in an algebraic structure —in your case, vector spaces.
Conjugations have many, many interrsting an useful aplications. The similarity of matrices is an special case of conjugation, and has the property that if $B=PXP^{-1}$ then $$B^n = P X^n P,$$ equality that is fundamental when calculating the exponential of a matrix (and hence, solving systems of differential equations).
