if there is a 3x3 matrix which has exactly 3 distinct eigenvalues why must it be diagonalizable?
basis such that is a diagonal matrix, where as usual is defined by
so suppose an matrix with entries in a field has distinct eigenvalues in , say let with then is a basis for
now consider the linear map defined by then and thus:
to see this we only need to show that are linearly independent. so suppose where not all of the coefficients are 0 and is minimal. call it (1). taking
of both sides of (1) gives us: and hence call this one (2). now multiply (1) by and subtract the result from (2) to get:
which contradicts the minimality of unless since are distinct, we must have
and thus by (1). contradiction! thus the elements of are linearly independent. Q.E.D.