Max number of Linearly Independent eigenvectors
I need help with this problem I came across. I give it in a simplified way:
The questions asks for the maximum number of linearly independent eigenvectors for the eigenvalue 7.
Now suppose the Matrix we are dealing with is in 3D and has eigenvectors:
, the k's are natural numbers (including zero).
Now the corresponding eigenvalues are:
What I thought at first was that since the expression has eigenvalue 3, adding one more term would mean adding something of the form , but the added expression would have eigenvalue 8. So at most we can have 1 eigenvector.
I'm not sure if just adding eigenvalues in the way I did is allowed.
I was thinking maybe the question meant the following:
Given the eigenvalue , if we equate this to seven, then we obtain , in which case there are a maximum of 6 linearly independent eigevectors:
Which is Right?