# Thread: Equivalence Relations

1. ## Equivalence Relations

I dont want the answer to this this but if someone could guide me in the right direction it would be much appreciated.

Consider the set M_n (R) of n x n matrices over R. For A,B \in M_n (R) say that B is similar to A if and only if there is an invertible matrix P with B = P^-1 A P. Show that similarity is an equivalence relation on M_n (R).

If i understand right to prove this i have to show that the set M_n (R) is refexive, symmetric and transitive but im not sure how to do this... Any help please!

Something weirds happened to my Latex stuff so im leaving it out, (R) means real numbers. \in is that symbol that looks like a mix between C and E.

2. Originally Posted by Deadstar
I dont want the answer to this this but if someone could guide me in the right direction it would be much appreciated.

Consider the set $\displaystyle M_n (R)$ of n x n matrices over R. For $\displaystyle A,B \in M_n (R)$ say that B is similar to A if and only if there is an invertible matrix P with $\displaystyle B = P^-1 A P$. Show that similarity is an equivalence relation on $\displaystyle M_n (R)$.

If i understand right to prove this i have to show that the set $\displaystyle M_n (R)$ is refexive, symmetric and transitive but im not sure how to do this... Any help please!

HOLD ON GOTTA FIX THE LATEX!
if a power has more than one character, you must put it in {}, that is type P^{-1} to get $\displaystyle P^{-1}$

for reflexivity, you must show that each matrix here is similar to itself, using $\displaystyle P = I_n$ will suffice. (sorry for giving you the answer, won't happen again )

for symmetry, you must show that if matrix $\displaystyle B$ is similar to matrix $\displaystyle A$, then matrix $\displaystyle A$ will be similar to matrix $\displaystyle B$ (just go by the definitions here. some ingenuity might be required to show that you can find such matrices $\displaystyle P$ for this to happen)

for transitivity, show that if $\displaystyle A$ is similar to $\displaystyle B$ and $\displaystyle B$ is similar to $\displaystyle C$, then $\displaystyle A$ will be similar to $\displaystyle C$. again, go by the definition of what it means to be similar, that is, the equation you were given

to get the symbol for the real numbers, type \mathbb{R}

example $$A,B \in M_n ( \mathbb{R} )$$ yields $\displaystyle A,B \in M_n ( \mathbb{R} )$

This is my 63th post!!!!

3. It is not $\displaystyle M_n (\Re )$ that you want to prove is reflexive, symmetric, and transitive but the relation.

Reflexive: Is every $\displaystyle A \in M_n \left( \Re \right)$ you must show that A is related to itself. Think identity, I.

Symmetric: If A is related to B, what about B to A? $\displaystyle B = P^{ - 1} AP \Rightarrow \quad PBP^{ - 1} = A \Rightarrow \quad \left( {P^{ - 1} } \right)^{ - 1} BP^{ - 1}$.

Transitive: Do you see how it works now?

4. Cheers for that ill see what i can do from there. Turns out the Latex was right (nearly) just my laptop went a bit mental and displayed it all over the page!