Finding the minimal polynomial of a matrix?

In summary, the minimal polynomial of a 3 x 3 matrix A with entries in Q, where char(A,x) = f(x) and f(x) is an irreducible polynomial cubic in Q, is m(x) = x^3 + (b/a)x^2 + (c/a)x + (d/a). This can be generalized to a degree n polynomial.
  • #1
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Homework Statement



Let f(x) be an irreducible polynomial cubic in Q. For example

f(x) = ax^3 + bx^2 + cx + d

Let A be a 3 x 3 matrix with entries in Q such that char(A,x) = f(x). Find the minimal polynomial m(x) of A. Can you generalize to a degree n polynomial?


Homework Equations





The Attempt at a Solution



If the char(A,x) = f(x) then the companion matrix is...

[0, 0, -d]
[1, 0, -c]
[0, 1, -b]

Since the companion matrix's characteristic polynomial = its minimal polynomial, does this mean the minimal polynomial is just f(x). I'm missing something, aren't I...
 
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  • #2
What about the a?
 
  • #3
Office_Shredder said:
What about the a?

Oh. The companion matrix would be...

[0 0 -d/a]
[1 0 -c/a]
[0 1 -b/a]

so m(x) = x^3 + (b/a)x^2 + (c/a)x + (d/a)?
 

1. What is a minimal polynomial of a matrix?

A minimal polynomial of a matrix is the monic polynomial of least degree that has the given matrix as a root. In other words, it is the smallest polynomial that, when evaluated at the matrix, gives the zero matrix.

2. How is the minimal polynomial of a matrix calculated?

The minimal polynomial of a matrix can be found by using the Cayley-Hamilton theorem, which states that every square matrix satisfies its own characteristic polynomial. By setting the characteristic polynomial equal to zero and solving for the variable, we can find the minimal polynomial.

3. Why is it important to find the minimal polynomial of a matrix?

The minimal polynomial of a matrix is important because it helps us understand the behavior of the matrix, such as its eigenvalues and eigenvectors. It also allows us to compute powers of the matrix efficiently, which is useful in many applications.

4. Can the minimal polynomial of a matrix have repeated roots?

Yes, the minimal polynomial of a matrix can have repeated roots. This means that the matrix can have fewer distinct eigenvalues than its size, and the minimal polynomial will reflect this by having repeated factors.

5. Does every matrix have a unique minimal polynomial?

Yes, every matrix has a unique minimal polynomial. This is because the minimal polynomial is determined by the matrix's characteristic polynomial, which is unique for each matrix.

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