Rational Canonical Form

[Maria00]

Hi, I'm just new here, I don't know if I'm on the right thread.:D

Homework Statement

Let F be a subfield of K. A, B be elements of Mn(F). Show that if A and B are similar over K, then A,B are similar over F. (Hint: what can be said about the rank of f(C(f(x)^m))^n? about the rank of f(C(f(x)^n))^m? where m<n are nonnegative integers and C(f(x)) is the companion matrix of the monic poly. f element of F[X].))

Homework Equations

The Attempt at a Solution

Here my attempt: Let A be an element of Mn(F) and mA(x) be the minimal poly. of A over F, Then it can be showed that mA(x) is also the minimal poly. of A over K. Since A & B are similar over K, then mA(x)=mB(x) where mB(X) is the min. poly of B over K. By similar argument, mB(x) is also the min. poly of B viewed as an element of Mn(F). Now, note that mA(x) is an element of F[x], then A is similar to it's rational canonical form (RCF). Also, B is similar to it's RCF. But mA(x)=mB(x)

=> RCF of A = RCF of B.

Hence, A & B are similar over F i.e, there exist a nonsingular Q element of Mn(F) s.t.
B=Q^-1AQ.

My problem is, i didn't use the hint. Thanks in advance.

 

[mighty2000]

Dear new member,

Welcome to the forum! I'm glad to see that you are already working on a problem and seeking help from others. This is a great place to discuss and learn about different scientific topics.

Now, let's take a look at the problem you have posted. The hint provided is actually quite useful in solving the problem. It suggests looking at the rank of certain matrices involving the companion matrix of a monic polynomial. Remember that the rank of a matrix is the maximum number of linearly independent rows or columns in that matrix.

One key property of the companion matrix is that its rank is equal to the degree of the monic polynomial it represents. This means that if we have a monic polynomial f(x) of degree n, the rank of the companion matrix C(f(x)) is also n.

Now, let's consider the matrices f(C(f(x))^m and f(C(f(x))^n, where m<n. Think about what these matrices represent and how they are related to the minimal polynomials of A and B. Can you see how the hint can be used to show that the rank of these matrices must be equal? And what does this tell us about the similarity of A and B over F?

I hope this helps! Keep working on the problem and don't hesitate to ask for clarification or further assistance. Good luck!