Linear Algebra

Section II. Similarity 363
3.19 Corollary An n×n matrix with n distinct eigenvalues is diagonalizable.
Proof. Form a basis of eigenvectors. Apply Corollary 2.4. QED
Exercises
3.20 For �each, find � the characteristic � � polynomial � and � the eigenvalues. � �
10 −9
1 2
0 3
0 0
(a)
(b)
(c)
(d)
4 −2
4 3
7 0
0 0
� �
1 0
(e)
0 1
� 3.21 For each matrix, find the characteristic equation, and the eigenvalues and
associated � eigenvectors. � � �
3 0
3 2
(a)
(b)
8 −1
−1 0
3.22 Find the characteristic equation, and the eigenvalues and associated eigenvectors
for this matrix. Hint. The eigenvalues are complex.
� �
−2 −1
5 2
3.23 Find the characteristic polynomial, the eigenvalues, and the associated eigenvectorsofthismatrix.
�
1 1
�
1
0 0 1
0 0 1
� 3.24 For each matrix, find the characteristic equation, and the eigenvalues and
associated eigenvectors.
� �
3 −2 0
�
0 1
�
0
(a) −2 3 0 (b) 0 0 1
0 0 5
4 −17 8
� 3.25 Let t: P2 →P2 be
a0 + a1x + a2x 2 ↦→ (5a0 +6a1 +2a2) − (a1 +8a2)x +(a0 − 2a2)x 2 .
Find its eigenvalues and the associated eigenvectors.
3.26 Find the eigenvalues and eigenvectors of this map t: M2 →M2.
� � � �
a b 2c a+ c
↦→
c d b − 2c d
� 3.27 Find the eigenvalues and associated eigenvectors of the differentiation operator
d/dx: P3 →P3.
3.28 Prove that the eigenvalues of a triangular matrix (upper or lower triangular)
are the entries on the diagonal.
� 3.29 Find the formula for the characteristic polynomial of a 2×2 matrix.
3.30 Prove that the characteristic polynomial of a transformation is well-defined.
� 3.31 (a) Can any non-�0 vector in any nontrivial vector space be a eigenvector?
That is, given a �v �= �0 from a nontrivial V , is there a transformation t: V → V
and a scalar λ ∈ R such that t(�v) =λ�v?
(b) Given a scalar λ, can any non-�0 vector in any nontrivial vector space be an
eigenvector associated with the eigenvalue λ?