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nomial is zero. In courses of linear algebra, you will study the necessary and
sufficient conditions for M to be diagonalizable.

In-Class Exercises
Pb. 8.13 Show that if M v = λ v , then M v = λ n v . That is, the eigen-
n
n
n
values of M are λ ; however, the eigenvectors v ‘s remain the same as those
of M.
Verify this theorem using the choice in Example 8.9 for the matrix M.
Pb. 8.14 Find the eigenvalues of the upper triangular matrix:

14/ 0  0
T =  −1 1 2/ 0 
 
 2 −3  1

Generalize your result to prove analytically that the eigenvalues of any trian-
gular matrix are its diagonal elements. (Hint: Use the previously derived
result in Pb. 8.1 for the expression of the determinant of a triangular matrix.)
Pb. 8.15 A general theorem, which will be proven to you in linear algebra
courses, states that if a matrix is diagonalizable, then, using the above notation:

–1
VDV = M
Verify this theorem for the matrix M of Example 8.9.
a. Using this theorem, show that:

∏
n
det(M = det( )D = λ i
)
i
b. Also show that:

–1
n
VD V = M n
5
c. Apply this theorem to compute the matrix M , for the matrix M of
Example 8.9.
Pb. 8.16 Find the non-zero eigenvalues of the 2 ⊗ 2 matrix A that satisfies
the equation:

A = A 3

© 2001 by CRC Press LLC