Minimal polynomials and invertibility

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In summary, the conversation discusses the relationship between the invertibility of g(T) and the greatest common divisor of the minimal polynomial of T and g(x). It is stated that g(T) is invertible if and only if the greatest common divisor of m(x) and g(x) is 1. The conversation also briefly mentions a server issue that occurred the day before.
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Treadstone 71
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Let [tex]T\in L(V)[/tex]. Let [tex]g(x)\in F[x][/tex] and let [tex]m(x)[/tex] be the minimal polynomial of [tex]T[/tex]. Show that [tex]g(T)[/tex] is invertible [tex]\Leftrightarrow[/tex] [tex]\gcd (m(x),g(x))=1[/tex].

Backwards is easy. For forwards, suppose I say that [tex]g(T)[/tex] is invertible implies that [tex]g(T)(v)=0 \Rightarrow v=0[/tex] and therefore [tex]g(x)[/tex] prime, therefore it is not divisible, and therefore [tex]\gcd (g,m)=1[/tex]. Is that correct?

Why is latex not showing up?
 
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Treadstone 71 said:
Let [tex]T\in L(V)[/tex]. Let [tex]g(x)\in F[x][/tex] and let [tex]m(x)[/tex] be the minimal polynomial of [tex]T[/tex]. Show that [tex]g(T)[/tex] is invertible [tex]\Leftrightarrow[/tex] [tex]\gcd (m(x),g(x))=1[/tex].

Backwards is easy. For forwards, suppose I say that [tex]g(T)[/tex] is invertible implies that [tex]g(T)(v)=0 \Rightarrow v=0[/tex] and therefore [tex]g(x)[/tex] prime, therefore it is not divisible, and therefore [tex]\gcd (g,m)=1[/tex]. Is that correct?

Why is latex not showing up?

They had some sort of server disaster yesterday, I take it.

-Dan
 

1. What is a minimal polynomial?

A minimal polynomial is the smallest degree monic polynomial that has the given number as a root. In other words, it is the polynomial of lowest degree that can be used to represent a given number.

2. How is a minimal polynomial related to invertibility?

A minimal polynomial is closely related to the invertibility of a square matrix. If a matrix has a minimal polynomial of degree n, then the matrix is invertible if and only if the coefficient of the highest degree term in the minimal polynomial is non-zero. In other words, if the minimal polynomial has a non-zero constant term, then the matrix is invertible.

3. Can a minimal polynomial have multiple roots?

Yes, a minimal polynomial can have multiple roots. In fact, if a minimal polynomial has a repeated root, then the matrix is not invertible. This is because a repeated root means that the matrix is not diagonalizable, which is a necessary condition for invertibility.

4. How can I find the minimal polynomial of a matrix?

To find the minimal polynomial of a matrix, you can use the Cayley-Hamilton theorem. This theorem states that every square matrix satisfies its own characteristic polynomial. So, to find the minimal polynomial, you can simply find the characteristic polynomial and then divide it by the leading coefficient to make it monic.

5. Can two matrices have the same minimal polynomial?

Yes, two matrices can have the same minimal polynomial. This is because the minimal polynomial is not unique and can be shared by multiple matrices. However, the matrices will not necessarily be similar, as the minimal polynomial only gives information about the diagonalizability and invertibility of a matrix, not its exact structure.

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