Cholesky decomposition on a symmetric matrix

Question

Let $S = S^T$ be a symmetric $n × n$ matrix with Cholesky decomposition $S = LDL^T$ . Prove or disprove the following statements:

The diagonal elements of D are the eigenvalues of S.

Any help will be much appreciated!

Answer 1

This is not true. Take $$ S=\pmatrix{4&2\\2&1} $$ with eigenvalues $0$ and $5$. The $LDL^T$ decomposition is given by $$ S=\pmatrix{ 1& 0 \\ 1/2& 1} \pmatrix{ 4 & 0 \\0&0} \pmatrix{ 1& 1/2 \\ 0& 1}. $$