## Diagonalization

Hi Sir,

I need a little help on diagonalising matrices. What do you do if you only have one eigenvalue. i.e. on question 1 of this weeks homework, the result of det(XI-A) is (X-2)^2=0 and (A-2I)v=0 gives x=y so my eigenvector is

(1)

(1)

How do I continue from here?

Thanks.

Reply:

Think about the following:

When is an nxn matrix with entries from a field F (say, Q, R ,C) diagonalizable over F?

(1) If and only if it has a basis of eigenvectors in F^n.

(2) If and only if the minimal polynomial has all roots in F and has no multiple roots.

If you consider the exercise from either of these perspectives, the answer should be clear.

MK

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