I have a fairly simple understading gap regarding the diagonlization process of a matrix which bothers me and ill be happy to receive an explanation regarding it or if perhaps I am mistaken then ofcourse correct me.
I understand the matrix digagonlization forumula - P⁻¹MP = D , what I dont understand is why we stop where D main diagonal λx,λy,λz....λn are each a different value in the main diagonal and we dont continue and build a formula where the final matrix is the identity matrix as we can P⁻¹MP × E1×E2...En elementry matrices get a final I identity matrix. P⁻¹MP × E1×E2...En = D× E1×E2...En = I Am I wrong ?
Yes, you are wrong. The goal of diagonalization is to find a diagonal matrix $D$ similar to the original matrix $M$. If one of the $\lambda_k$'s is is different from $1$, then $M$ is not similar to the identity matrix. By the way, the only matrix similar to the identity matrix is the identity matrix itself.