Diagonalized matrices take the form 
where the matrix 
represents a set of eigenvectors (a basis for eigenspace) and 
is a diagonal matrix (zeros everywhere but the diagonal) containing the eigenvalues. For example, given that 
and 
has eigenvalues 
and 
with eigenvectors 
and 
respectively, we find the following diagonalization for 
: 
Diagonalizaed matrices are useful for many things such as raising a matrix to a power: 
where the matrix represents a set of eigenvectors (a basis for eigenspace) and is a diagonal matrix (zeros everywhere but the diagonal) containing the eigenvalues. For example, given that and has eigenvalues and with eigenvectors and respectively, we find the following diagonalization for :