A.7.2 Triangular Matrices
Although it is usually hard to solve the eigenvalue problem, there is a kind of matrix for which this is not
the case. These are the upper or lower triangular matrices. I will illustrate by examples.
Example A.7.3 Let A =
. Find its eigenvalues.
You need to solve
Thus the eigenvalues are just the diagonal entries of the original matrix. You can see it would work this
way with any such matrix. These matrices are called upper triangular.
Stated precisely, a matrix A
upper triangular if Aij
= 0 for all i > j.
Similarly, it is easy to find the eigenvalues for a lower triangular
matrix, one which has all zeros above the main diagonal.