Similarity transformation for set of matrices represent the same group [closed]
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I am trying to solve the following problem, but I do not really know where to even start... If someone could help, I'd be very grateful.
For a set of matrices $S(g)$ forming a representation of group $G$, with $g in G$, the similarity transformation of $S(g)$, i.e. $S'(g) = U S(g) U^{-1}$ is also a representation of $G$.
matrices group-theory
asked Nov 23 at 21:15
bjorn
1104
closed as off-topic by Jean-Claude Arbaut, amWhy, Leucippus, Cesareo, Shailesh Nov 24 at 3:36
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Jean-Claude Arbaut, amWhy, Leucippus, Cesareo, Shailesh
If this question can be reworded to fit the rules in the help center, please edit the question.
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I am trying to solve the following problem, but I do not really know where to even start... If someone could help, I'd be very grateful.
For a set of matrices $S(g)$ forming a representation of group $G$, with $g in G$, the similarity transformation of $S(g)$, i.e. $S'(g) = U S(g) U^{-1}$ is also a representation of $G$.
matrices group-theory
asked Nov 23 at 21:15
bjorn
1104
closed as off-topic by Jean-Claude Arbaut, amWhy, Leucippus, Cesareo, Shailesh Nov 24 at 3:36
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Jean-Claude Arbaut, amWhy, Leucippus, Cesareo, Shailesh
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I am trying to solve the following problem, but I do not really know where to even start... If someone could help, I'd be very grateful.
For a set of matrices $S(g)$ forming a representation of group $G$, with $g in G$, the similarity transformation of $S(g)$, i.e. $S'(g) = U S(g) U^{-1}$ is also a representation of $G$.
matrices group-theory
asked Nov 23 at 21:15
bjorn
1104
I am trying to solve the following problem, but I do not really know where to even start... If someone could help, I'd be very grateful.
For a set of matrices $S(g)$ forming a representation of group $G$, with $g in G$, the similarity transformation of $S(g)$, i.e. $S'(g) = U S(g) U^{-1}$ is also a representation of $G$.
matrices group-theory
matrices group-theory
asked Nov 23 at 21:15
bjorn
1104
asked Nov 23 at 21:15
bjorn
1104
asked Nov 23 at 21:15
bjorn
1104
asked Nov 23 at 21:15
bjorn
1104
asked Nov 23 at 21:15
bjorn
1104
1104
closed as off-topic by Jean-Claude Arbaut, amWhy, Leucippus, Cesareo, Shailesh Nov 24 at 3:36
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Jean-Claude Arbaut, amWhy, Leucippus, Cesareo, Shailesh
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Jean-Claude Arbaut, amWhy, Leucippus, Cesareo, Shailesh Nov 24 at 3:36
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Jean-Claude Arbaut, amWhy, Leucippus, Cesareo, Shailesh
If this question can be reworded to fit the rules in the help center, please edit the question.
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1 Answer
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Hint: You need to show that $S'$ is also a homomorphism.
So check if $S'(gh)=S'(g)S'(h),,forall g,hin G$. This should be pretty straightforward, using that $S$ is a homomorphism, i.e. $S(gh)=S(g)S(h)$.
answered Nov 23 at 21:26
Chris Custer
9,3653624
Ah, maybe I understand. So, $S'(gh) = U S(gh) U^{-1} = U S(g) S(h) U^{-1} = US(g)U^{-1} US(h)U^{-1} = S'(g)(S'(h)$, which holds for all $g,h in G$, and $gh in G$ from the closure axiom?
– bjorn
Nov 23 at 21:36
@bjorn You got it.
– Chris Custer
Nov 23 at 21:37
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
Hint: You need to show that $S'$ is also a homomorphism.
So check if $S'(gh)=S'(g)S'(h),,forall g,hin G$. This should be pretty straightforward, using that $S$ is a homomorphism, i.e. $S(gh)=S(g)S(h)$.
answered Nov 23 at 21:26
Chris Custer
9,3653624
Ah, maybe I understand. So, $S'(gh) = U S(gh) U^{-1} = U S(g) S(h) U^{-1} = US(g)U^{-1} US(h)U^{-1} = S'(g)(S'(h)$, which holds for all $g,h in G$, and $gh in G$ from the closure axiom?
– bjorn
Nov 23 at 21:36
@bjorn You got it.
– Chris Custer
Nov 23 at 21:37
add a comment |
up vote
1
down vote
accepted
Hint: You need to show that $S'$ is also a homomorphism.
So check if $S'(gh)=S'(g)S'(h),,forall g,hin G$. This should be pretty straightforward, using that $S$ is a homomorphism, i.e. $S(gh)=S(g)S(h)$.
answered Nov 23 at 21:26
Chris Custer
9,3653624
Ah, maybe I understand. So, $S'(gh) = U S(gh) U^{-1} = U S(g) S(h) U^{-1} = US(g)U^{-1} US(h)U^{-1} = S'(g)(S'(h)$, which holds for all $g,h in G$, and $gh in G$ from the closure axiom?
– bjorn
Nov 23 at 21:36
@bjorn You got it.
– Chris Custer
Nov 23 at 21:37
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Hint: You need to show that $S'$ is also a homomorphism.
So check if $S'(gh)=S'(g)S'(h),,forall g,hin G$. This should be pretty straightforward, using that $S$ is a homomorphism, i.e. $S(gh)=S(g)S(h)$.
answered Nov 23 at 21:26
Chris Custer
9,3653624
Hint: You need to show that $S'$ is also a homomorphism.
So check if $S'(gh)=S'(g)S'(h),,forall g,hin G$. This should be pretty straightforward, using that $S$ is a homomorphism, i.e. $S(gh)=S(g)S(h)$.
answered Nov 23 at 21:26
Chris Custer
9,3653624
answered Nov 23 at 21:26
Chris Custer
9,3653624
answered Nov 23 at 21:26
Chris Custer
9,3653624
answered Nov 23 at 21:26
Chris Custer
9,3653624
9,3653624
Ah, maybe I understand. So, $S'(gh) = U S(gh) U^{-1} = U S(g) S(h) U^{-1} = US(g)U^{-1} US(h)U^{-1} = S'(g)(S'(h)$, which holds for all $g,h in G$, and $gh in G$ from the closure axiom?
– bjorn
Nov 23 at 21:36
@bjorn You got it.
– Chris Custer
Nov 23 at 21:37
add a comment |
Ah, maybe I understand. So, $S'(gh) = U S(gh) U^{-1} = U S(g) S(h) U^{-1} = US(g)U^{-1} US(h)U^{-1} = S'(g)(S'(h)$, which holds for all $g,h in G$, and $gh in G$ from the closure axiom?
– bjorn
Nov 23 at 21:36
@bjorn You got it.
– Chris Custer
Nov 23 at 21:37
Ah, maybe I understand. So, $S'(gh) = U S(gh) U^{-1} = U S(g) S(h) U^{-1} = US(g)U^{-1} US(h)U^{-1} = S'(g)(S'(h)$, which holds for all $g,h in G$, and $gh in G$ from the closure axiom?
– bjorn
Nov 23 at 21:36
Ah, maybe I understand. So, $S'(gh) = U S(gh) U^{-1} = U S(g) S(h) U^{-1} = US(g)U^{-1} US(h)U^{-1} = S'(g)(S'(h)$, which holds for all $g,h in G$, and $gh in G$ from the closure axiom?
– bjorn
Nov 23 at 21:36
@bjorn You got it.
– Chris Custer
Nov 23 at 21:37
@bjorn You got it.
– Chris Custer
Nov 23 at 21:37
add a comment |
