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$M_n(F)$ represents the collection of all square matrices of size $n \times n$ with elements from the field $F$. Let's take a specific matrix $A$ from $M_n(\mathbb{R})$, where $\mathbb{R}$ is the set of real numbers. We'll define a set $C(A)$ as the set of matrices $B$ in $M_n(\mathbb{R})$ such that the product $AB$ is equal to $BA$: $$C(A)=\{B\in M_n(\mathbb{R})\mid AB=BA\}$$

It can be shown relatively easily that the set $C(A)$ forms a vector space over the field of real numbers, $\mathbb{R}$. We are tasked with demonstrating that the dimension of this vector space is greater than or equal to $n$: $$\text{dim }C(A)\geq n$$

I've extensively researched various approaches to this problem and came across several methods that involve concepts like the Jordan canonical form, the minimal polynomial, and the characteristic polynomial of matrix A, ideas related to linear transformations. However, I aim to tackle this problem using the knowledge I've acquired in my course. Some of them are:

  • Matrix elementary operations and elementary matrices.
  • Row-reduced echelon form of a matrix.
  • Vector space, including its definition, properties, basis, dimension, kernel, and nullity.
  • Subspaces and their properties, as well as their relationship to the vector spaces they are part of.
  • Understanding linear independence and dependence of vectors.
  • The concept of a coordinate vector relative to a given basis.
  • Recognition of row and column spaces of a matrix, although I haven't yet encountered the rank of a matrix.
  • $\dots$

Because the methods we have for solving the problem are quite basic, I believe it's enough to demonstrate that we have a minimum of $n$ elements in the basis. I suspect that techniques involving diagonal matrices, nullity, and elementary matrices might be useful for this purpose.

I would greatly appreciate any assistance or hints to help me solve this problem using the knowledge and concepts I've already been taught in my course. If you'd like to utilize advanced ideas, try to demonstrate them using fundamental concepts or establish their characteristics using basic tools. For instance, if you plan to utilize the Jordan canonical form, begin by explaining what it is and then highlight the specific characteristics of this concept that you will use to address the main problem. However, in this context, simply mentioning the names of the relevant concepts along with their properties or referencing a source that introduces and proves these properties is sufficient and greatly appreciated.

I have carefully read all the comments, and I am extremely grateful to those who took the time to comment. However, their responses were beyond the scope of what we can incorporate or apply.

EDIT

Please check this solution.

Mason Rashford
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    Consider polynomials in $A$. If the minimal polynomial of $A$ is $m_A$, then $\dim F[A] = \deg m_A$. This proves the result for matrices with $m_A = \chi_A$. Perhaps you can now use some kind of continuity argument regarding the map $A\mapsto C(A)$. – Pedro Nov 05 '23 at 15:20
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    https://math.stackexchange.com/questions/1379878/cm-a-in-m-n-mathbbc-mid-am-ma-is-a-subspace-of-dimension-at-least?rq=1 – QuantumSuperfield Nov 05 '23 at 15:38
  • See also my post here – Ben Grossmann Nov 05 '23 at 15:45
  • @QuantumSuperfield I don't know about Jordan canonical form and I want to solve this problem in a more elementary way without using JCF. – Mason Rashford Nov 05 '23 at 15:51
  • @BenGrossmann I don't know about Jordan canonical form and I want to solve this problem in a more elementary way without using JCF. – Mason Rashford Nov 05 '23 at 15:52
  • @Pedro What do you mean by polynomials in $A$? – Mason Rashford Nov 05 '23 at 15:53
  • @DietrichBurde My professor asked me to solve it without using JCF unfortunately. – Mason Rashford Nov 05 '23 at 15:56
  • Is this really written in the homework? If this was only told you, I know by experience that this is often a miscommunication. On the other hand, the first comment gives you an idea how to do it without JNF. – Dietrich Burde Nov 05 '23 at 15:58
  • @DietrichBurde He didn't teach JCF and we cannot use concepts that are not taught yet. – Mason Rashford Nov 05 '23 at 16:08
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    what text are you using? JCF is definitely not needed here. E.g. if you had done ex 4.6.18 in Artin's Algebra 1st edition, you would know the answer. That problem is to consider $T:\mathbb C^{n \times n}\longrightarrow \mathbb C^{n \times n}$ given by $T(B) = AB- BA$ and show $\text{rank}\big(T\big) \leq n^2 -n$ so the result follows by rank-nullityy. JCF and minimal polynomials don't show up for another 8 chapters but Artin expects you to solve this for diagonalizable $A$ first then use analytic techniques to get the general answer. – user8675309 Nov 05 '23 at 16:45
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    @user8675309 I'm reading Hoffman and Kunze book. – Mason Rashford Nov 05 '23 at 17:06
  • @user8675309 I'm also not allowed to use linear transformations :( – Mason Rashford Nov 05 '23 at 17:44
  • @Mason What decompositions are you allowed to use? Can we use the "primary" or "direct sum" decomposition? – Ben Grossmann Nov 05 '23 at 17:58
  • @BenGrossmann They are not also taught yet. If you want to use something new, you should prove the properties of it that you're using during the proof using concepts we know already. – Mason Rashford Nov 05 '23 at 18:34
  • @user8675309 I suspect that the lower semicontinuity of rank is beyond the scope of the asker's class – Ben Grossmann Nov 05 '23 at 20:59
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    Your list of concepts contains pretty much the entirety of the first two chapters of Hoffman and Kunze, and yet you don't know linear trnasformation, which is literally the topic of Chapter 3? Also, the problem doesn't exist anywhere in the first two chapters of Hoffmann and Kunze, either. – Divide1918 Nov 06 '23 at 13:04
  • @Divide1918 Yes, my professor taught first two chapters and gave us this problem. We are not allowed to use concepts beyond chapter 1 and 2. – Mason Rashford Nov 06 '23 at 13:10
  • @Mason I strongly recommend that you ask your professor for a hint. I'm a bit doubtful that your professor has a suitable solution in mind. – Ben Grossmann Nov 06 '23 at 19:22

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