[Math] LDU matrix decomposition

Question

Let $A$ be a matrix that can be written as $LDU$ for some lower unitriangular matrix $L$, some diagonal matrix $D$ and some upper unitriangular matrix $U$.
Then, are the eigenvalues of $A$ the same as those of $D$? Are the eigenvalues of $A$ the elements on the diagonal of $D$?

Answer 1

Not necessarily. Consider this example: $$ \pmatrix{ 1&0\\ 1&1 } \pmatrix{ 2&0\\ 0&3 } \pmatrix{ 1&1\\ 0&1 } = \pmatrix{ 2&2\\ 2&5 } $$ which has eigenvalues $1$ and $6$.