Thread: Polynomial of Matrix

Let A,B belong to Mn and AB=BA. Let p(•) be any polynomial, say of degree m. Show that p(A)p(B) = p(B)p(A).

The only thing that I have in my notes states that
pA(t)=(t-1)(t-2)...(t-n).

I have no idea even how to begin this. Any help would be great. Thanks!

Originally Posted by page929

Let A,B belong to Mn and AB=BA. Let p(•) be any polynomial, say of degree m. Show that p(A)p(B) = p(B)p(A).

The only thing that I have in my notes states that
pA(t)=(t-1)(t-2)...(t-n).

I have no idea even how to begin this. Any help would be great. Thanks!

p(•)=

What is ? ? ? ?

If the polynomials can be written as



Then I would use the discrete convolution formula and try to figure out the math. Be careful with the indices, it seems to me you have to play a little bit with those (I haven't done the computations but this seems the way to go).

Originally Posted by Also sprach Zarathustra

p(•)=

What is ? ? ? ?

Here is what I came up with...can you let me know if I am right?

actually, what you have written down is p(A), p(B), p(AB), and p(BA).

p(A)p(B) is the product of two polynomials, and will be of degree 2m. and not every coefficient will be a perfect square.

for example, if p(x) was x^2 + 1, p(A)p(B) = (A^2 + I)(B^2 + I) = A^2B^2 + A^2 + B^2 + I = (AB)^2 + A^2 + B^2 + I, not (AB)^2 + I.

the formula is messy, because you have for each term of degree k, several things to sum together. for example,

the next term after wil be , and the term after that is:

.