Triangular matrix/Characteristic polynomial/Eigenvalues/Example

For an upper triangular matrix

${}M={\begin{pmatrix}d_{1}&\ast &\cdots &\cdots &\ast \\0&d_{2}&\ast &\cdots &\ast \\\vdots &\ddots &\ddots &\ddots &\vdots \\0&\cdots &0&d_{n-1}&\ast \\0&\cdots &\cdots &0&d_{n}\end{pmatrix}}\,,$

the characteristic polynomial is

${}\chi _{M}=(X-d_{1})(X-d_{2})\cdots (X-d_{n})\,,$

due to fact. In this case, we have directly a factorization of the characteristic polynomial into linear factors, so that we can see immediately the zeroes and the eigenvalues of ${}M$, namely just the diagonal elements ${}d_{1},d_{2},\ldots ,d_{n}$ (which might not be all different).