Characteristic and minimal polynomials

In summary, the conversation discusses the equivalence of various statements regarding a finite dimensional complex vector space V and its linear operator T. These statements include V having a basis consisting of eigenvectors of T, T being represented by a diagonal matrix, all eigenvalues of T having multiplicity one, and the characteristic polynomial of T being equal to the minimal polynomial of T. The conversation also raises the question of whether B implies C, but it is concluded that it does not necessarily imply it, as the two "multiplicities" (geometric and algebraic) do not have to be equal for T to be represented by a diagonal matrix. However, it is suggested that there may be a missing assumption in the problem statement.
  • #1
corey2014
22
0

Homework Statement


Let V be a finite dimensional complex vector space and T be the linear operator of V. Prove that the following are equivalent

a V has a basis consisting of eigenvectors of T.
b T can be represented by a diagonal matrix.
c all the eigenvalues of T have multiplicity one.
d. the Characteristic polynomial of T equals the minimal polynomial of T.


Homework Equations



Not really applicable

The Attempt at a Solution



Ok so I proved A implies B, However, I feel that B does not imply C.

I just want to see if my argument is valid. because the identity matrix, is definitely diaganol however, it does not have a multiplicity of One. Can I assume wlog that T has distinct eigenvalues along the main diaganol?
 
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  • #2
You are quite right. But I wouldn't assume wlog. That looses generality. B doesn't imply C period. I think whoever wrote the problem up had a memory lapse or forgot to state an assumption. You could assume they forgot that assumption and proceed from there.
 
Last edited:
  • #3
corey2014 said:

Homework Statement


Let V be a finite dimensional complex vector space and T be the linear operator of V. Prove that the following are equivalent

a V has a basis consisting of eigenvectors of T.
b T can be represented by a diagonal matrix.
c all the eigenvalues of T have multiplicity one.
? There are two "multiplicities", the geometric multiplicity (number of independent eigenvectors corresponding to the eigenvalue) and algebraic multiplicity (multiplicity as a root of the characteristic polynomial). But neither of those must be one in order that T can be represented by a diagonal matrix, only that, for each eigenvalue, the two multiplicities be the same.

d. the Characteristic polynomial of T equals the minimal polynomial of T.


Homework Equations



Not really applicable

The Attempt at a Solution



Ok so I proved A implies B, However, I feel that B does not imply C.

I just want to see if my argument is valid. because the identity matrix, is definitely diaganol however, it does not have a multiplicity of One. Can I assume wlog that T has distinct eigenvalues along the main diaganol?
 

1. What is the characteristic polynomial of a matrix?

The characteristic polynomial of a matrix is a polynomial function that is derived from the matrix itself. It is used to find the eigenvalues of the matrix, which are the values that, when multiplied by the identity matrix and subtracted from the original matrix, result in a zero matrix.

2. How is the characteristic polynomial related to the eigenvalues of a matrix?

The roots of the characteristic polynomial are the eigenvalues of the matrix. This means that by finding the roots of the characteristic polynomial, one can determine the eigenvalues of the matrix.

3. What is the minimal polynomial of a matrix?

The minimal polynomial of a matrix is the monic polynomial of lowest degree that, when applied to the matrix, results in a zero matrix. It is used to find the Jordan canonical form of a matrix.

4. How is the minimal polynomial related to the characteristic polynomial?

The minimal polynomial is a factor of the characteristic polynomial. This means that the characteristic polynomial can be factored into the minimal polynomial and some other polynomial, where the minimal polynomial is the highest common factor.

5. Can a matrix have multiple characteristic and minimal polynomials?

Yes, a matrix can have multiple characteristic and minimal polynomials. This is because the characteristic polynomial is dependent on the choice of basis for the matrix, and the minimal polynomial can change if the matrix is similar to another matrix. However, the eigenvalues and Jordan canonical form will remain the same.

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