$mathbb{S} = {MX - XM mid X in mathbb{R}^{ntimes n}}$ gives $dim mathbb{S} leq n^{2} - n$ [on hold]












0














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







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put on hold as off-topic by anomaly, KReiser, Leucippus, user91500, José Carlos Santos 2 days ago


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  • "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

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  • Well, what do you think about the problem?
    – anomaly
    2 days ago










  • Still thinking.
    – BasicUser
    2 days ago
















0














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







share|cite|improve this 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














0












0








0







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







share|cite|improve this question















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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 2 days ago









Henning Makholm

238k16303540




238k16303540










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


















  • 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










1 Answer
1






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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.






share|cite|improve this answer




























    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    0














    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.






    share|cite|improve this answer


























      0














      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.






      share|cite|improve this answer
























        0












        0








        0






        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.






        share|cite|improve this answer












        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.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 2 days ago









        OmnomnomnomOmnomnomnom

        127k788176




        127k788176















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