Talk:Matrix similarity

message: Examples would be useful of permutation-similar and unitarily equivalent matrices, plus a pair that is neither of those but is still similar. 128.83.138.141 (talk) 16:55, 29 January 2010 (UTC)
thread_title:  Examples 

message: I came to this article looking for an answer to the question
: Is there a procedure to determine if two arbitrary matrices are similar? More strongly, given two matrices A and B, is it possible to find P such that B = P^{-1}AP?
Obviously if B is diagonalizable then diagonalization will find A and P, but I'm wondering about the general case for arbitrary A and B. Could this information be added to the article? — Unbitwise (talk) 15:21, 23 April 2010 (UTC)
replies: text: Just find Jordan canonical form of each matrix and check that they're the same. -- X7q (talk) 16:07, 23 April 2010 (UTC)

text: And btw this is already in the article: "By looking at the Jordan forms or rational canonical forms of A and B, one can immediately decide whether A and B are similar". -- X7q (talk) 16:12, 23 April 2010 (UTC)
thread_title:  Determining similarity 

message: The title "similar matrix" is incorrect. A matrix can not be similar. Two of more matrices may be similar.82.75.67.221 (talk) 22:48, 23 January 2009 (UTC)
replies: text: I don't think so; what if I take a matrix A and look at the set of all matrices similar to A. I would call all of those matrices "similar matrices", and I would call one in particular a "similar matrix". Suppose you had matrix trading cards. Fred says, "I have a matrix such-and-such!" Then Joe overhears and says "Oh, I have a similar matrix". I rest my case. --Fusionshrimp 128.194.39.250 (talk) 18:56, 4 April 2009 (UTC)
replies: text: It looks rather silly to speak of (a) similar matrix, as if a single matrix may be called similar. Nijdam (talk) 18:20, 1 September 2011 (UTC)

text: Why not: "similarity (matrix)"? Nijdam (talk) 11:09, 6 August 2012 (UTC)

text: I second the proposal to the name change. There is no "similar matrix" after all; unlike, say, orthogonal matrix. similarity (matrix) sounds good to me. -- Taku (talk) 12:17, 6 August 2012 (UTC)
thread_title:  Title 

message: If T = P^{-1}SP then what is S as a function of T? Just granpa (talk) 21:33, 16 May 2019 (UTC)
replies: text: If T = P^{-1}SP then S = PTP^{-1}.—Anita5192 (talk) 22:33, 16 May 2019 (UTC)
thread_title:  What is S as a function of T? 

message: Look at what wolfgram says  I also think P^{-1} is used to change the basis from {e_i} to {p_i}. So we would have x^' = P^{-1}x and y^' = P^{-1}y, in this new basis we can use the simpler rotation matrix. y^' = Ax^' So this means,
P^{-1}y = AP^{-1}x 
y = Tx = PAP^{-1}x
I think what I have written above is right since Py^' = y means that while the scalar values in y can be used with the standard basis vectors to find the vector y. The scalar values in y^' is used to choose a linear combination of the columns of P which is the same as y. So y^' is y written in the language of P. As 3b1b clearly puts it here the matrix P constructed with the new basis vectors as columns multiplied by a vector x^' in that language brings it over to our e_i language, i.e. we get x.  — Preceding unsigned comment added by Aditya8795 (talk • contribs) 16:47, 30 November 2019 (UTC)
replies: text: The two definitions are equivalent, as \exists P\mid  PAP^{-1}=B and \exists Q\mid  Q^{-1}AQ=B are equivalent assertions; it suffices to set Q=P^{-1}. So, the choice of the side where the exponent –1 appears is arbitrary. D.Lazard (talk) 17:55, 30 November 2019 (UTC)
thread_title:  This might be wrong, some expert needs to look into it.