Linear algebra find the minimal polynomial

In summary, the conversation is about finding the possibilities for the minimal polynomial of the linear operator M: V -> V, where M^2 + 1_v = 0. The attempt at a solution involves rewriting the operator as M^3 + M + I_v using M^2 = -1_v, and guessing a possible minimal polynomial of x^3-x-1. However, the correct minimal polynomial is M^3 = (-M).
  • #1
catsarebad
72
0

Homework Statement


M: V -> V linear operator st M^2 + 1_v = 0
find the POSSIBILITIES for min. pol. of M^3+2M^2+M+3I_v


Homework Equations





The Attempt at a Solution



using M^2 = -1_v,
i rewrote the operator(?) as
M^3 + M + I_v

i don't know what to do. i guessed min poly to be like x^3-x-1
but what would be other possibilities. i seriously doubt the one i guessed is even correct.

i know that, if min poly is u(x) then u(M) = 0.
 
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  • #2
catsarebad said:

Homework Statement


M: V -> V linear operator st M^2 + 1_v = 0
find the POSSIBILITIES for min. pol. of M^3+2M^2+M+3I_v


Homework Equations





The Attempt at a Solution



using M^2 = -1_v,
i rewrote the operator(?) as
M^3 + M + I_v

i don't know what to do. i guessed min poly to be like x^3-x-1
but what would be other possibilities. i seriously doubt the one i guessed is even correct.

i know that, if min poly is u(x) then u(M) = 0.

M^3=(-M), isn't it?
 
  • #3
got it.
 

FAQ: Linear algebra find the minimal polynomial

1. What is a minimal polynomial in linear algebra?

A minimal polynomial is the smallest degree polynomial that has the given matrix as its root. It is used to find the eigenvalues of a matrix and can also be used to determine the diagonalizability of a matrix.

2. How do you find the minimal polynomial of a matrix?

The minimal polynomial can be found by first finding the characteristic polynomial of the matrix. Then, factor the characteristic polynomial and find the smallest degree polynomial that has all the same roots as the characteristic polynomial. This polynomial is the minimal polynomial.

3. What is the purpose of finding the minimal polynomial?

Finding the minimal polynomial allows us to determine the eigenvalues of a matrix and also to determine whether a matrix is diagonalizable. It is an important tool in linear algebra and is used in various applications such as solving systems of differential equations and computing powers of a matrix.

4. Can a matrix have more than one minimal polynomial?

Yes, a matrix can have multiple minimal polynomials. This can happen if the matrix has repeated eigenvalues. In this case, there will be multiple polynomials of different degrees that have the matrix as its root.

5. How does the degree of the minimal polynomial relate to the size of the matrix?

The degree of the minimal polynomial is always less than or equal to the size of the matrix. In some cases, the degree of the minimal polynomial may be equal to the size of the matrix, but it will never be greater.

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