Thread: question about similar matrices

To prove 2 matrices are similar you need to check:

1. same trace
2. same rank
3. same.......
4. same eigen values
5. same algebraic and geometric count (i dunno what it's called in english...)

My question is can 2 matrices have the same eigenvalues and algebraic and geometric count, but not have the same trace or rank... as in not similar?

Yes. Two matrices are similar if and only if they have the same eigenvalues and the same eigenvectors. Saying that they have the eigenvaues as well as same algebraic degrees (multiplicity of eigenvalue as a solution to the characteristic equation) and geometric degrees (dimension of eigenspace for each eigenvalue) is necessary but not sufficient.

Originally Posted by jayshizwiz

To prove 2 matrices are similar you need to check:

1. same trace
2. same rank
3. same.......
4. same eigen values
5. same algebraic and geometric count (i dunno what it's called in english...)

My question is can 2 matrices have the same eigenvalues and algebraic and geometric count, but not have the same trace or rank... as in not similar?

Just to emphasize, two matrices can have all the described properties above and not be similar.