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I might be wrong, but it seems to me that there are two notions of "simmilarity" with regards to matrices which are slightly different:

  • A is similar to B if an invertible Matrix P exists s.t. $A = PBP^{-1}$

In the context of diagonalizable matrices though, where a matrix A is diagonalizable if it similar to a diagonal matrix, it is no longer the case that B would be the diagonal matrix (which A is similar too as above), but the whole complex "$PBP^{-1}$" is a diagonal matrix, i.e. A is similar to $PBP^{-1}$?

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    $\begingroup$ It's hard for me to follow what is the problem here: the definition of similarity of matrices is the one you give in the first point. Period. Now, there's this special kind of matrices that are called "diagonalizable" if they're similar to a diagonal matrix. Period again...so? $\endgroup$ – DonAntonio Mar 23 '13 at 20:14
  • $\begingroup$ But for this special kind the definition is applied differently - they are not, written like I did, similar to a diagonal matirx but equal to one. Do you see the difference I mean? $\endgroup$ – user62487 Mar 23 '13 at 20:17
  • $\begingroup$ @user62487 no they are not equal. $B$ is the diagonal matrix and $P B P^{-1}$ is just the matrix $A$. $\endgroup$ – Dominic Michaelis Mar 23 '13 at 20:18
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    $\begingroup$ I really don't, @user62487 ... $\endgroup$ – DonAntonio Mar 23 '13 at 20:20
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    $\begingroup$ @user62487 nope $P^{-1} A P$ is the diagonal matrix $\endgroup$ – Dominic Michaelis Mar 23 '13 at 20:25
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As $A=P B P^{-1}$ it is for sure similar to $P B P^{-1}$ with $I P B P^{-1} I^{-1}$ with the identiy matrix $I$.

Your second statement says that it is similar to a diagonal matrix $B$ such that $A=P B P^{-1}$

Multiplying $P$ from right and $P^{-1}$ from left yields $$B= P^{-1} A P$$ which is the diagonal matrix

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