University of Oxford mathematician Dr Tom Crawford goes through a full proof of the Spectral Theorem. Check out ProPrep with a 30-day free trial to see how it can help you to improve your performance in STEM-based subjects: https://www.proprep.uk/info/TOM-Crawford

The video goes through a full proof of the Spectral Theorem, which states that every real, symmetric matrix, has real eigenvalues, and can be diagonalised using a basis of its eigenvectors. The first part of the proof uses the eigenvalue equation to show that any eigenvalue is in fact equal to its complex conjugate, and thus is real. The second part of the proof shows that a matrix similarity transformation using an orthogonal matrix exists, and results in a diagonal matrix. We first construct an orthonormal basis (where the first vector is an eigenvector) using the Gram-Schmidt process, and then use these vectors as the columns of our orthogonal matrix. Next, we show that the resulting similarity matrix is also symmetric. This then allows us to conclude that the first row and first column are diagonal as required. The final step is to use induction on the size of the matrix. Assuming the result is true for a (n-1) x (n-1) matrix, we use our earlier calculation to construct the final orthogonal matrix, and show that when it is used as a change of basis matrix the result is diagonal, as we wanted.