Change of Basis + Geometric, Algebraic Multiplicities

[psholtz]

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.

 

[tiny-tim]

hi psholtz!

… 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!