1. ## Diagonalizable

I am trying to prove that the charactertic polynomial of a diagonalizable matrix will "split".

I know intuitively this has to be true because a diagonalizable matrix would be similar to a diagonal one, which would have the same charactristic polynomial... and obviously that polynomial will split since its determinant will just be determined by the diagonal entries.. but I am stuck on how to formally prove it.

Thank you in advance for any tips!!!

2. What do you mean by "split"?

3. ## Split

split meaning factors... so the polynomial f(t) splits over the field if there are scalars $c, a_1, a_2, ..., a_n$ in F such that $f(t)=c(t-a_1)(t-a_2)...(t-a_n)$

4. Originally Posted by MatthewD
split meaning factors... so the polynomial f(t) splits over the field if there are scalars $c, a_1, a_2, ..., a_n$ in F such that $f(t)=c(t-a_1)(t-a_2)...(t-a_n)$
So the characteristic polynomial