254 CHAPTER 12. SPECTRAL THEORY
and the row reduced echelon form is 1 0 0 00 1 −1 00 0 0 0
and so the eigenvectors are of the form t
(0 1 1
)Twhere t ̸= 0.
Finally find the eigenvectors for λ = 4. The augmented matrix for the system of equa-tions needed to find these eigenvectors is −2 2 −2 | 0
1 −1 −1 | 0−1 1 −3 | 0
and the row reduced echelon form is 1 −1 0 0
0 0 1 00 0 0 0
.
Therefore, the eigenvectors are of the form t(
1 1 0)T
where t ̸= 0.
12.1.4 Triangular MatricesAlthough it is usually hard to solve the eigenvalue problem, there is a kind of matrix forwhich this is not the case. These are the upper or lower triangular matrices. I will illustrateby examples.
Example 12.1.8 Let A =
1 2 40 4 70 0 6
. Find its eigenvalues.
You need to solve
0 = det
1 2 4
0 4 70 0 6
−λ
1 0 00 1 00 0 1
= det
1−λ 2 40 4−λ 70 0 6−λ
= (1−λ )(4−λ )(6−λ ) .
Thus the eigenvalues are just the diagonal entries of the original matrix. You can see itwould work this way with any such matrix. These matrices are called upper triangular.Stated precisely, a matrix A is upper triangular if Ai j = 0 for all i > j. Similarly, it is easyto find the eigenvalues for a lower triangular matrix, on which has all zeros above the maindiagonal.