Linear Algebra, 2020a

Section II. Similarity 421
a criterion for diagonalizability. (While it is useful for the theory, note that in
applications finding eigenvalues this way is typically impractical; for one thing
the matrix may be large and finding roots of large-degree polynomials is hard.)
In the next section we study matrices that cannot be diagonalized.
Exercises
3.22 This matrix has two eigenvalues λ 1 = 3, λ 2 =−4.
( ) 4 1
−8 −5
Give two different diagonal form matrices with which it is similar.
3.23 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
̌ 3.24 For each matrix, find the characteristic equation and the eigenvalues and
associated
(
eigenvectors.
) ( )
3 0
3 2
(a)
(b)
8 −1 −1 0
3.25 Find the characteristic equation, and the eigenvalues and associated eigenvectors
for this matrix. Hint. The eigenvalues are complex.
( ) −2 −1
5 2
3.26 Find the characteristic polynomial, the eigenvalues, and the associated eigenvectors
of this matrix.
⎛
1 1
⎞
1
⎝0 0 1⎠
0 0 1
̌ 3.27 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
3.28 For each matrix, find the characteristic polynomial, and the eigenvalues and associated
eigenspaces. Also find the algebraic and geometric multiplicities.
⎛
⎞ ⎛ ⎞
( ) 1 3 −3
2 3 −3
13 −4
(a)
(b) ⎝−3 7 −3⎠
(c) ⎝0 2 −3⎠
−4 7
−6 6 −2
0 0 1
̌ 3.29 Let t: P 2 → P 2 be this linear map.
a 0 + a 1 x + a 2 x 2 ↦→ (5a 0 + 6a 1 + 2a 2 )−(a 1 + 8a 2 )x +(a 0 − 2a 2 )x 2
Find its eigenvalues and the associated eigenvectors.
3.30 Find the eigenvalues and eigenvectors of this map t: M 2 → M 2 .
( ) ( )
a b 2c a + c
↦→
c d b − 2c d