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Let's say we have a given matrix ##G##. I want to find a set of ##M## matrices so that ##MG = GM## and prove that this is a group. How can I approach this problem?
By more conditions, e.g. ##M=0## is possible in your setup. Do you look for an addition group?Let's say we have a given matrix ##G##. I want to find a set of ##M## matrices so that ##MG = GM## and prove that this is a group. How can I approach this problem?
I'm not sure we're on the same page. I've been learning linear algebra for one month only, so I might not see your point immidiately. My teacher mentioned this problem and I found it interesting.##\{\,M\,|\,MG=GM\,\}## is already a group, say ##H##. Then ##G\in H## and ##H## is Abelian. But I have the strong feeling that you will not be satisfied.
Tell me. My dictionary gave me aggresziv or eröszakos.Aggressive element is an element that makes every other element equal to itself. Like multiplying real numbers by zero. Not sure if aggressive element is the right term in English, I learnt it in Hungarian.
Easy: Let ##H:=\{\,M\in \mathbb{R}^{n \times n}\,|\,MG=GM\,\}##.The matrices are in ##\mathbb{R}^{n \times n}##. And as ##G## is given, I would like to express what the ##M## matrices must look like.
I'm not sure we're on the same page. I've been learning linear algebra for one month only, so I might not see your point immidiately. My teacher mentioned this problem and I found it interesting.
##M,N \in H \Longrightarrow M+N \in H##The picture I have in mind is that a group is an algebraic structure which means that there is a set and a binary operation with the following properties
- closure for that operation
##M+(N+P)=(M+N)+P##- associativity
##M \in H \Longrightarrow -M \in H##- there is an inverse element for every element in the set
##0\in H##- there is an identity element in the set
This is a rather boring solution, but if we must not invert the matrices, the more interesting multiplicative case isn't possible. The entire question including "aggressive" reminds me of algebras which are used in biology.So far, my problem is that I don't really see how the statement "matrices commuting with a given matrix form a group" fits into this picture. What is the set, what is the operation, etc.
It's aggresszív in Hungarian. I didn't find anything for agressive element, how do you call this in English?Tell me. My dictionary gave me aggresziv or eröszakos.
But how is this a proof that these rules also apply to commuting matrices too? Maybe it should be trivial but I don't see how the sum of two matrices that commute with ##G## also commutes with ##G##. Same for the other conditions.Easy: Let ##H:=\{\,M\in \mathbb{R}^{n \times n}\,|\,MG=GM\,\}##.
##M,N \in H \Longrightarrow M+N \in H##
##M+(N+P)=(M+N)+P##
##M \in H \Longrightarrow -M \in H##
##0\in H##
I think it's an "absorbing element". https://en.wikipedia.org/wiki/Absorbing_elementIt's aggresszív in Hungarian. I didn't find anything for agressive element, how do you call this in English?
If you interpret "is a group" to mean "is a group under the operation of multiplication" then the problem asks for a proof of a false statement. Begin by fixing the problem.Let's say we have a given matrix ##G##. I want to find a set of ##M## matrices so that ##MG = GM## and prove that this is a group. How can I approach this problem?
##G(A+B) = ? = ? = (A+B)G##Maybe it should be trivial but I don't see how the sum of two matrices that commute with G" role="presentation">G also commutes with G" role="presentation">G
Let ##G## be a ##n \times n## matrixLet's say we have a given matrix ##G##. I want to find a set of matrices ##M## so that ##MG = GM## and prove that this is a group. How can I approach this problem?
Doesn't every element in a group have an inverse? If Det(G)=0 , then G is not invertible.Let ##G## be a ##n \times n## matrix
Define ##S## to be the set of ##n \times n## matrices with nonzero determinant: ##S = \{ n \times n \text{ matrix } A | \text{ } det(A) \neq 0 \}##
Define ##T## to be the set of ##n \times n## matrices with nonzero determinant that commute with ##G##: ##T = \{M \in S | \text{ } MG=GM \}##
It is easy to show that ##T## is a group
Questions:
Is it necessary that ##G \in S## ?. That is, is it necessary that ##det(G) \neq 0## ?
Does ##G## belong to ##T##, i.e., ##G \in T## ?
Commutator, but even this depends on the structure: ##[G,M]=GM-MG## or ##[G,M]=GMG^{-1}M^{-1}.##If your matrix G is the identity, it will commute with non-invertible matrices, and these will not be invertible. I saw a related name commutant?
If ##\operatorname{det}G =0## then ##-G## is still the inverseDoesn't every element in a group have an inverse? If Det(G)=0 , then G is not invertible.
Is this the same as Centralizer?https://en.wikipedia.org/wiki/Centralizer_and_normalizer (See the first few lines, where it is also nammed commutant)Commutator, but even this depends on the structure: ##[G,M]=GM-MG## or ##[G,M]=GMG^{-1}M^{-1}.##
But I thought this was a multiplicative group. An additive group has every element invertible, doesn't it, by -G itself, right?If ##\operatorname{det}G =0## then ##-G## is still the inverse
Obviously, ##T## is a multiplicative group, consisting of ##n \times n## matrices with nonzero determinant.Let ##G## be a ##n \times n## matrix
Define ##S## to be the set of ##n \times n## matrices with nonzero determinant: ##S = \{ n \times n \text{ matrix } A | \text{ } det(A) \neq 0 \}##
Define ##T## to be the set of ##n \times n## matrices with nonzero determinant that commute with ##G##: ##T = \{M \in S | \text{ } MG=GM \}##
It is easy to show that ##T## is a group
Questions:
Is it necessary that ##G \in S## ?. That is, is it necessary that ##det(G) \neq 0## ?
Does ##G## belong to ##T##, i.e., ##G \in T## ?
That was one of the difficulties with the OP. It hasn't been specified, and to automatically assume invertible matrices if he talks about ##\mathbb{R}^{n\times n}## is a bit of a stretch.But I thought this was a multiplicative group. An additive group has every element invertible, doesn't it, by -G itself, right?
Obviously, this is an unsupported assumption.Obviously, ##T## is a multiplicative group, consisting of ##n \times n## matrices with nonzero determinant.
But if you have an aditive group then, at least in my understanding, everything commutes.That was one of the difficulties with the OP. It hasn't been specified, and to automatically assume invertible matrices if he talks about ##\mathbb{R}^{n\times n}## is a bit of a stretch.
No one said it won't be a boring solution. At least it is one which doesn't depend on assumptions made by others.But if you have an aditive group then, at least in my understanding, everything commutes.