Change of Basis + Geometric, Algebraic Multiplicities

In summary, changing the basis of a matrix representation of a linear operator will not affect the eigenvalues, but it will not change the geometric multiplicities of those eigenvalues. The ability to diagonalize a linear operator depends on its canonical representation, and a change of basis will not alter the geometric multiplicities.
  • #1
psholtz
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Making a change of basis in the matrix representation of a linear operator will not change the eigenvalues of that linear operator, but could making such a change of basis affect the geometric multiplicities of those eigenvalues?

I'm thinking that the answer is "no", it cannot..

Since if it did, it would affect/change the ability to diagonalize the linear operator, and any given linear operator is going to have only one canonical representation..

But, I just wanted to make sure.
 
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  • #2
hi psholtz! :wink:
psholtz said:
… But, I just wanted to make sure.

fair enough!

no, the geometric multiplicities of eigenvalues depend on the dimension of a subspace of the vector space, and no change of basis is going to alter that! :smile:
 

Related to Change of Basis + Geometric, Algebraic Multiplicities

1. What is a change of basis?

A change of basis is a mathematical process used in linear algebra to represent the same vector or matrix in different coordinate systems. It involves finding a new set of basis vectors that can be used to describe the original vector or matrix in a different way.

2. How is a change of basis related to geometric and algebraic multiplicities?

The geometric and algebraic multiplicities of an eigenvalue (or root) of a matrix or linear transformation are important factors in determining the change of basis. The geometric multiplicity refers to the number of linearly independent eigenvectors associated with an eigenvalue, while the algebraic multiplicity refers to the number of times the eigenvalue appears as a root of the characteristic polynomial. These multiplicities can help determine the dimension of the new basis and the transformations needed to change the original basis.

3. What is the significance of geometric and algebraic multiplicities?

Geometric and algebraic multiplicities can provide insight into the behavior and properties of a matrix or linear transformation. They can help determine the dimension of the vector space, the number of distinct eigenvalues, and the types of transformations that can be performed on the space.

4. How do you calculate geometric and algebraic multiplicities?

To calculate the geometric multiplicity, one can find the null space of the matrix or linear transformation associated with a specific eigenvalue. The dimension of this null space is the geometric multiplicity. To calculate the algebraic multiplicity, one can find the degree of the corresponding root in the characteristic polynomial.

5. Can geometric and algebraic multiplicities be the same?

Yes, it is possible for geometric and algebraic multiplicities to be the same. This occurs when all the eigenvectors associated with a particular eigenvalue are linearly independent, making the geometric multiplicity equal to the algebraic multiplicity.

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