if there is a 3x3 matrix which has exactly 3 distinct eigenvalues why must it be diagonalizable?
well, it depends how much you know about diagonalizable matrices. i'll assume you know nothing but this definition that an matrix with entries in a field is diagonalizable iff has a
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.