Diagonal matrix Definition and 55 Threads

Homework Statement Suppose the matrix A , B are diagonalizable and have the same eigenvectors. Show AB=BA Homework Equations The Attempt at a Solution There exists a matrix P s.t. (P^-1)AP=(P^-1)BP I played around with this, and could not get anywhere..

Perhaps a silly question. I have a vector: a=[a_1 \ a_2 \ a_3 \ ...\ a_n]^T that I want to turn into a diagonal matrix. Is there an elegant way to represent this? I thought maybe something like: a^TI would do, but it doesn't. I suppose I can use the kronecker delta and subscript...

I have been thinking about this for a week and I simply can't get anywhere. consider a 4x4 diagonal matrix such that if a11 > a22 > a33 > a44, (0's everywhere else) then it is similar to another matrix with the same diagonal but 0's in upper triangular part an any numbers in the lower...

is the inverse of a diagonal matrix always just calculated by taking the inverses of each number in the matrix?

diagonal matrix is a matrix with each element is0 excep for elements on the major diagonal. so, does it mean that ( 2 0 0 0 0 0 0 0 3 ) is not a diagonal matrix??