Exercise 1

The main idea is to select one eigenvector \(\ket{\lambda_N}\) and consider the \(N-1\) dimensional subspace spanned by linearly independent vectors \(v_i\), \(i=1, \dots, N-1\) orthogonal to \(\ket{\lambda_N}\): \[\braket {v_i | \lambda_N} = 0\]

Since the operator \(M\) is Hermitian, the images \(M \ket{v_i}\) of all vectors of this subspace are again members of this subspace and orthogonal to \(\ket{\lambda_N}\): \[ \braket{v_i | M \ket{\lambda_N}} = \braket {v_i | \lambda_N \ket {\lambda_N}} = \braket{\bra{v_i}M | \ket{\lambda_N} } = 0 \]

The operator \(M\), restricted to this subspace, has at least one eigenvector \(\ket{\lambda_{N-1}}\) which is orthogonal to \(\ket{\lambda_N}\). Repeat this procedure until the dimension is \(1\) and you found \(N\) orthogonal eigenvectors. Use the Gram-Schmidt procedure to make this set orthonormal.

A rigorous proof ist not easy, even though the task gives the impression that it is.

I won’t copy a complete proof here. The reader who is more interested in this topic will find complete information searching for the spectral theorem and there are different types of proofs which can be found in the Web or in math textbooks.

If you are looking for a textbook on linear algebra (as a companion textbook to the TTM QM), I would recommend the excellent and very popular book from Sheldon Axler: Linear Algebra Done Right. It has a Creative Commons BY-NC license and is legally available without cost in PDF or Kindle format.