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.