Eigenvalues of a Triangular Matrix

Write out $\det (A-\lambda I)$ for an upper triangular matrix.

The determinant of a triangular matrix is the product of its diagonal entries.

1. 1. Form the characteristic matrix

   $A-\lambda I$ is also upper triangular, with diagonal entries $a_{ii}-\lambda$.

2. 2. Take the determinant

   The determinant of an upper triangular matrix is the product of its diagonal entries, so the characteristic polynomial is ${\prod }_i(a_{ii}-\lambda )$, whose roots are exactly the diagonal entries $a_{ii}$.

Answer: Its diagonal entries.