Linear Algebra, 2020a

Section II. Similarity 411
Exercises
̌ 2.6 Repeat Example 2.5 for the matrix from Example 2.2.
̌ 2.7 Diagonalize this matrix by following the steps of Example 2.5.
( ) 1 1
0 0
(a) Set up the matrix-vector equation described in Lemma 2.4 and rewrite it as a
linear system.
(b) By considering solutions for that system, find two vectors to make a basis.
(Consider separately the system in the x = 0 and x ≠ 0 cases. Also, recall that
the zero vector cannot be a member of a basis.)
(c) Use that basis in the similarity diagram to get the diagonal matrix as the
product of three others.
̌ 2.8 Follow Example 2.5 to diagonalize this matrix.
( ) 0 1
1 0
(a) Set up the matrix-vector equation and rewrite it as a linear system.
(b) By considering solutions for that system, find two vectors to make a basis.
(Consider separately the x = 0 and x ≠ 0 cases. Also, recall that the zero vector
cannot be a member of a basis.)
(c) With that basis use the similarity diagram to get the diagonalization as the
product of three matrices.
2.9 Diagonalize
( )
these upper
(
triangular
)
matrices.
−2 1 5 4
(a)
(b)
0 2 0 1
2.10 If we try to diagonalize the matrix of Example 2.3
( ) 0 0
N =
1 0
using the method of Example 2.5 then what goes wrong?
(a) Draw the similarity diagram with N.
(b) Set up the matrix-vector equation described in Lemma 2.4 and rewrite it as a
linear system.
(c) By considering solutions for that system, find the trouble. (Consider separately
the x = 0 and x ≠ 0 cases.)
̌ 2.11 What form do the powers of a diagonal matrix have?
2.12 Give two same-sized diagonal matrices that are not similar. Must any two
different diagonal matrices come from different similarity classes?
2.13 Give a nonsingular diagonal matrix. Can a diagonal matrix ever be singular?
̌ 2.14 Show that the inverse of a diagonal matrix is the diagonal of the inverses, if no
element on that diagonal is zero. What happens when a diagonal entry is zero?
2.15 The equation ending Example 2.5
( ) −1 ( )( )
1 1 3 2 1 1
=
0 −1 0 1 0 −1
( ) 3 0
0 1