Cyclic Decomposition Theorem

[Kindayr]

I took an Intermediate Linear Algebra course all last year (two semesters worth) and we covered the CDT. My professor didn't teach it well, and I got my first B- in university because of it (didn't affect my GPA but still irritating).

I didn't understand a lot of the canonical form stuff because I just straight up had a horrible prof who didn't allow us to grow a good conceptual idea of what the hell was happening.

Then today i was reading the beginning of Artin's "Algebra" on groups, its it slapped me in the face and I was awestruck. He was talking about permutations, and the cyclic groups that may come out of them. My mind was blown as soon as I started thinking of the CDT.

So, I'm just hoping for someone to explain it, and hopefully it matches my intuition.

(I may be confusing CDT and RCF, I'm tired and can't think straight, that's why I'm waiting for someone first, unless they ask me for my explanation, then I will explain it tomorrow when I'm rejuvenated)

 

[Kindayr]

Alright now that I have time to type. This won't be over the top rigourous, just trying to get my ideas onto 'paper'.

Let $V$ be a vector space over a field $\mathbb F$ and let $T$ be a linear operator whose characteristic polynomial factors in $k$ irreducibles over $\mathbb F$

Basically, my intuition now tells me that we can find a basis so that the matrix representation of $T$ with respect to this new basis is 'prettier' and easier to work with. What I think the CDT tells us is that V can be broken down into $k$ $T$-invariant subspaces that are disjoint. These subspaces are the null-spaces of each individual irreducible with $T$ substituted in. The basis we want is the union of the bases of these $k$ subspaces.

I feel this, if this is correct (see how badly i was taught it hahahah), is analogous to the subsets a permutation creates. For example, let a permutation $P:12345\to45132$. Then we get the cyclic subsets $(143)(25)$.

Is this the proper intuition to have? Or am I way off the mark?

 

[micromass]

Alright now that I have time to type. This won't be over the top rigourous, just trying to get my ideas onto 'paper'.

Let $V$ be a vector space over a field $\mathbb F$ and let $T$ be a linear operator whose characteristic polynomial factors in $k$ irreducibles over $\mathbb F$

Basically, my intuition now tells me that we can find a basis so that the matrix representation of $T$ with respect to this new basis is 'prettier' and easier to work with. What I think the CDT tells us is that V can be broken down into $k$ $T$-invariant subspaces that are disjoint. These subspaces are the null-spaces of each individual irreducible with $T$ substituted in. The basis we want is the union of the bases of these $k$ subspaces.

No, I'm afraid this is not correct. Something that you describe is in fact the primary decomposition theorem (which is weaker than the cyclic decomposition theorem). For the primary decomposition theorem, we factor the minimal polynomial as

$$p=p_1^{r_1}...p_n^{r_n}$$

where the $p_i$are distinct. Then we indeed take the nullspace $W_i$ of $p_i(T)^{r_i}$. Then we will indeed have $V=W_1\oplus ... + W_n$.

However, the cyclic decomposition theorem is much harder and I don't think that such a trick will work there.

 

[Kindayr]

Ooooooooh that makes sense.

I don't currently have my Friedberg on me, so I wasn't able to look up the actual definitions.

Well I'm definitely retaking that course because I really feel I missed out on a lot of good things. Ughzors.

Thank you though!

 

[blue_raver22]

The Cyclic Decomposition Theorem is a fundamental result in linear algebra that allows us to break down a matrix into simpler, more easily understood components. It states that any matrix can be decomposed into a direct sum of cyclic subspaces, where each subspace is generated by a single vector that is repeatedly acted upon by the matrix.

This theorem has applications in various areas of mathematics, including group theory and abstract algebra. It allows us to understand the structure of a matrix and its corresponding linear transformation in terms of simpler, cyclic components.

It is unfortunate that your previous professor did not teach this concept well, as it is a crucial topic in linear algebra. However, it is great that you have now found a better understanding of it through your own reading and exploration of the topic.

If you need further clarification or explanation of the Cyclic Decomposition Theorem, I am happy to help. It is important to have a strong understanding of this theorem in order to fully grasp the concepts of linear algebra and its applications. I am glad that you have found a renewed interest in this topic and I hope it continues to inspire you in your studies.