For an
upper triangular matrix
-
![{\displaystyle {}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}}\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b96a76aef0569c43b8f6d76f4426c8d408048ce)
the
characteristic polynomial
is
-
![{\displaystyle {}\chi _{M}=(X-d_{1})(X-d_{2})\cdots (X-d_{n})\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5372a6df42334c5cc79f0831df5dfb2c67d5b34a)
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
, namely just the diagonal elements
(which might not be all different).