$mathbb{S} = {MX - XM mid X in mathbb{R}^{ntimes n}}$ gives $dim mathbb{S} leq n^{2} - n$ [on hold]
Fix $Minmathbb{R}^{ntimes n}$ and define the vector space
begin{align*}
mathbb{S} = left{MX - XM mid X in mathbb{R}^{ntimes n}right}.
end{align*}
Show that $dim mathbb{S} leq n^{2} - n$.
linear-algebra matrices vector-spaces
edited 2 days ago
Henning Makholm
238k16303540
asked 2 days ago
BasicUserBasicUser
15914
put on hold as off-topic by anomaly, KReiser, Leucippus, user91500, José Carlos Santos 2 days ago
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 provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – anomaly, KReiser, Leucippus, user91500, José Carlos Santos
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
Fix $Minmathbb{R}^{ntimes n}$ and define the vector space
begin{align*}
mathbb{S} = left{MX - XM mid X in mathbb{R}^{ntimes n}right}.
end{align*}
Show that $dim mathbb{S} leq n^{2} - n$.
linear-algebra matrices vector-spaces
edited 2 days ago
Henning Makholm
238k16303540
asked 2 days ago
BasicUserBasicUser
15914
put on hold as off-topic by anomaly, KReiser, Leucippus, user91500, José Carlos Santos 2 days ago
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 provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – anomaly, KReiser, Leucippus, user91500, José Carlos Santos
If this question can be reworded to fit the rules in the help center, please edit the question.
Well, what do you think about the problem?
– anomaly
2 days ago
Still thinking.
– BasicUser
2 days ago
add a comment |
Fix $Minmathbb{R}^{ntimes n}$ and define the vector space
begin{align*}
mathbb{S} = left{MX - XM mid X in mathbb{R}^{ntimes n}right}.
end{align*}
Show that $dim mathbb{S} leq n^{2} - n$.
linear-algebra matrices vector-spaces
edited 2 days ago
Henning Makholm
238k16303540
asked 2 days ago
BasicUserBasicUser
15914
Fix $Minmathbb{R}^{ntimes n}$ and define the vector space
begin{align*}
mathbb{S} = left{MX - XM mid X in mathbb{R}^{ntimes n}right}.
end{align*}
Show that $dim mathbb{S} leq n^{2} - n$.
linear-algebra matrices vector-spaces
linear-algebra matrices vector-spaces
edited 2 days ago
Henning Makholm
238k16303540
asked 2 days ago
BasicUserBasicUser
15914
edited 2 days ago
Henning Makholm
238k16303540
asked 2 days ago
BasicUserBasicUser
15914
edited 2 days ago
Henning Makholm
238k16303540
edited 2 days ago
Henning Makholm
238k16303540
edited 2 days ago
Henning Makholm
238k16303540
238k16303540
asked 2 days ago
BasicUserBasicUser
15914
asked 2 days ago
BasicUserBasicUser
15914
asked 2 days ago
BasicUserBasicUser
15914
15914
put on hold as off-topic by anomaly, KReiser, Leucippus, user91500, José Carlos Santos 2 days ago
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 provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – anomaly, KReiser, Leucippus, user91500, José Carlos Santos
If this question can be reworded to fit the rules in the help center, please edit the question.
put on hold as off-topic by anomaly, KReiser, Leucippus, user91500, José Carlos Santos 2 days ago
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 provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – anomaly, KReiser, Leucippus, user91500, José Carlos Santos
If this question can be reworded to fit the rules in the help center, please edit the question.
Well, what do you think about the problem?
– anomaly
2 days ago
Still thinking.
– BasicUser
2 days ago
add a comment |
Well, what do you think about the problem?
– anomaly
2 days ago
Still thinking.
– BasicUser
2 days ago
Well, what do you think about the problem?
– anomaly
2 days ago
Well, what do you think about the problem?
– anomaly
2 days ago
Still thinking.
– BasicUser
2 days ago
Still thinking.
– BasicUser
2 days ago
add a comment |
1 Answer
1
active
oldest
votes
Sketch of proof: Define the linear map $T:Bbb C^{n times n} to Bbb C^{n times n}$ by $T(X) = MX - XM$. We see that $ker(T) = {X mid MX = XM}$. By this post or this post, we see that $dim ker (T) geq n$. By the rank-nullity theorem, we have $$
dim(Bbb S) = dim(operatorname{im(T)}) = n^2 - dim ker(T) leq n^2-n
$$
as desired.
answered 2 days ago
OmnomnomnomOmnomnomnom
127k788176
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
Sketch of proof: Define the linear map $T:Bbb C^{n times n} to Bbb C^{n times n}$ by $T(X) = MX - XM$. We see that $ker(T) = {X mid MX = XM}$. By this post or this post, we see that $dim ker (T) geq n$. By the rank-nullity theorem, we have $$
dim(Bbb S) = dim(operatorname{im(T)}) = n^2 - dim ker(T) leq n^2-n
$$
as desired.
answered 2 days ago
OmnomnomnomOmnomnomnom
127k788176
add a comment |
Sketch of proof: Define the linear map $T:Bbb C^{n times n} to Bbb C^{n times n}$ by $T(X) = MX - XM$. We see that $ker(T) = {X mid MX = XM}$. By this post or this post, we see that $dim ker (T) geq n$. By the rank-nullity theorem, we have $$
dim(Bbb S) = dim(operatorname{im(T)}) = n^2 - dim ker(T) leq n^2-n
$$
as desired.
answered 2 days ago
OmnomnomnomOmnomnomnom
127k788176
add a comment |
Sketch of proof: Define the linear map $T:Bbb C^{n times n} to Bbb C^{n times n}$ by $T(X) = MX - XM$. We see that $ker(T) = {X mid MX = XM}$. By this post or this post, we see that $dim ker (T) geq n$. By the rank-nullity theorem, we have $$
dim(Bbb S) = dim(operatorname{im(T)}) = n^2 - dim ker(T) leq n^2-n
$$
as desired.
answered 2 days ago
OmnomnomnomOmnomnomnom
127k788176
Sketch of proof: Define the linear map $T:Bbb C^{n times n} to Bbb C^{n times n}$ by $T(X) = MX - XM$. We see that $ker(T) = {X mid MX = XM}$. By this post or this post, we see that $dim ker (T) geq n$. By the rank-nullity theorem, we have $$
dim(Bbb S) = dim(operatorname{im(T)}) = n^2 - dim ker(T) leq n^2-n
$$
as desired.
answered 2 days ago
OmnomnomnomOmnomnomnom
127k788176
answered 2 days ago
OmnomnomnomOmnomnomnom
127k788176
answered 2 days ago
OmnomnomnomOmnomnomnom
127k788176
answered 2 days ago
OmnomnomnomOmnomnomnom
127k788176
127k788176
add a comment |
add a comment |
Well, what do you think about the problem?
– anomaly
2 days ago
Still thinking.
– BasicUser
2 days ago