I have this:
A=any square matrix over any field F
p(X)=any polynomial over F
P=any invertible matrix over F
B=inverse(P)AP=P^(-1)AP
I need to show that p(B)=inverse(P)p(A)P=P^(-1)p(A)P.
How would I prove this?
Sorry about that.
It used the important property of similar matricies.
.
Let the polynomial F[x] be,
.
Then,
And,
Use the property mentioned about,
You can use left-right distributive laws in a matrix,
But,
Thus,
(The only mistake with you question is that you said a polynomial over a field. That is not what I assumed).