Lecture Notes in Differential Equations - Bruce E. Shapiro

125
Simplifying
4x − 2 + 3z = 0 (14.62)
x + 3 + z = 0 (14.63)
x + 3 + z = 0 (14.64)
The second and third equations are now the same because we have fixed
one of the values. The remaining two equations give two equations in two
unknowns:
4x + 3z = 2 (14.65)
x + z = −3 (14.66)
The solution is x = 11, z = −14. Therefore an eigenvalue of A corresponding
to λ = −2 is v = (11, 1, −14), as is any constant multiple of this vector.
Definition 14.31. The main diagonal of a square matrix A is the list
(a 11 , a 22 , . . . , a nn ).
Definition 14.32. A diagonal matrix is a square matrix that only has
non-zero entries on the main diagonal.
Theorem 14.33. The eigenvalues of a diagonal matrix are the elements
of the diagonal.
Definition 14.34. An upper (lower) triangular matrix is a square
matrix that only has nonzero entries on or above (below) the main diagonal.
Theorem 14.35. The eigenvalues of an upper (lower) triangular matrix
line on the main diagonal.