What are similarity invariants for companion matrix?

[catsarebad]

[SOLVED] what are similarity invariants for companion matrix?

here is how companion matrix is defined Companion matrix - Wikipedia, the free encyclopedia

similarity invariant i think is basically the unique roots of characteristic polynomial. like (x-p1) (x-p2) (x-p3) etc.

the first thing that jumps to my mind is trying to do A-rI and trying to write out character poly as (x-p1) (x-p2) etc but that seems like a nightmare.

edit: NVM DONE.

 

[jvicens]

Hello,

Thank you for bringing up this interesting question about similarity invariants for companion matrices. I can provide some additional insights and information on this topic.

Firstly, let's define what a companion matrix is. It is a square matrix that represents a linear transformation on a vector space. It has a specific structure, where the first row contains the coefficients of the characteristic polynomial of the linear transformation, and the remaining rows are constructed in a way that makes it a nilpotent matrix. This means that the matrix raised to a certain power will result in a zero matrix.

Now, let's discuss the similarity invariants for companion matrices. Similarity invariants are properties of a matrix that are preserved under similarity transformations. In other words, if two matrices are similar, they will have the same similarity invariants. For companion matrices, the main similarity invariant is the eigenvalues or roots of the characteristic polynomial.

This is because, as you mentioned, the characteristic polynomial of a companion matrix can be written as (x-p1)(x-p2)(x-p3)...(x-pn), where p1, p2, p3,..., pn are the eigenvalues of the matrix. This means that the eigenvalues are unique and do not change under similarity transformations. Therefore, they are a strong similarity invariant for companion matrices.

Another similarity invariant for companion matrices is the determinant. The determinant of a companion matrix is equal to the product of its eigenvalues. This means that the determinant is also a unique property of the matrix that does not change under similarity transformations.

In addition to these, there are other properties that are preserved under similarity transformations, such as the trace, rank, and characteristic polynomial itself. All of these can be used as similarity invariants for companion matrices.

I hope this helps to clarify the concept of similarity invariants for companion matrices. If you have any further questions or would like to discuss this topic further, please let me know.