I have the Transformation T s.t:
and
Find a basis B of R3 such that M_B(T), the B-matrix of T, is diagonal.
I really dont know how to approach this i know i want to find
P so that but how do i find A etC?
Let
Let then .
Let . If this basis diagnolizes then it means is a diagnol matrix.
Let be the transition matrix from to then it means where is coordinate vector with respect to and is coordinate vector with respect to .
And .
Now find eigenvalues of .
Then it would mean would diagnolize .
Now this means
If you evaluate this equality at you get
Thus, we get where is first colomn of . It follows that and that gives you what is. Now do the same procedure for .
I started to do what you said and computed the eigenvectors and im assuming it cant be right as the eigenvalues and vectors are probably the most hideous thing ive ever seen in my life ( done on maple). The answer that the book tells me is very simply and they seem to just get A out of no where.
edit: nm i got it thanks i was thinking stupidly.