So, I was reading the book Nielsen and Chuang and it's introductory chapter on Quantum Mechanics and it had a theorem called the "Spectral Decomposition Theorem" which states that an Operator $M$ is a normal operator if and only if it can be diagonalized in the orthonormal basis (which turn out to be the eigen-vectors).
Now, I have some trouble understanding the forward proof, which was to prove that if I have a Normal operator, it can be diagonalized. I am attaching the proof that is written in the book. Ultimately, it uses Projectors onto eigenspaces of a particular eigen value $P_\lambda=\sum_i |\lambda ; i\rangle \langle\lambda;i |$ and it's Orthogonal complement $Q_\lambda \equiv I-P_\lambda$ and rewrite $M=(P_\lambda+Q_\lambda)M(P_\lambda+Q_\lambda)$ which then simplifies to $M=P_\lambda MP_\lambda +Q_\lambda M Q_\lambda$ and using principle of mathematical induction, one can indeed prove that M can be diagonalized with respect to some orthonormal basis.
The part of the proof that I don't understand is that somehow, it implies that the operator $M$ can be diagonalized with respect to eigenvectors of this Normal Operator and that they are necessarily orthogonal. Also, if I try to write down $M$ in outer-product representation, it somehow simplifies as $M=\sum_i \lambda_i |e_i\rangle\langle e_i|$ where $\lambda_i$'s are the eigen-values and $|e_i\rangle$'s are the eigenvectors. I have literally no idea how this decomposition was implied by the above arguments. Any sort of help in the understanding of this is appreciated.
P.S.- I would really appreciate if you could explain the arguments with respect to the way that it is given in the proof in this book.