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Maria00
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Hi, I'm just new here, I don't know if I'm on the right thread.:D
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].))
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.
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.