Question: Orthoganalized Eigenvectors and Diagonalisation

Good Morning everyone and happy easter

I have a short question on Maples Eigenvectors procedure. I want to diagonalize a Matrix A, which works works perfectly using the following code:

A:= some quadratic (complex) float matrix

(l,L):=Eigenvectors(A):

B:=L . DiagonalMatrix(l) . MatrixInverse(L):

Excecuting this code tells me, that A=B, so it works.

Yet, what I would like to do now, is to convert the Eigenvector-matrix L into a matrix K, where the eigenvectors are orthogonalized to each other (Gram-Schmidt) and I still get

B = K . DiagonalMatrix(l) . MatrixInverse(K):

Is this possible or am I making a serious mistake, considering that L and l are related ( since Vector L[i] is eigenvector of eigenvalue l[i]  ), but K is not to l ?

Thank you very much for any suggestions

Karl