Linear Algebra, 2020a

412 Chapter Five. Similarity
is a bit jarring because for P we must take the first matrix, which is shown as an
inverse, and for P −1 we take the inverse of the first matrix, so that the two −1
powers cancel and this matrix is shown without a superscript −1.
(a) Check that this nicer-appearing equation holds.
( ) ( )( )( ) −1
3 0 1 1 3 2 1 1
=
0 1 0 −1 0 1 0 −1
(b) Is the previous item a coincidence? Or can we always switch the P and the
P −1 ?
2.16 Show that the P used to diagonalize in Example 2.5 is not unique.
2.17 Find a formula for the powers of this matrix. Hint: see Exercise 11.
( ) −3 1
−4 2
2.18 We can ask how diagonalization interacts with the matrix operations. Assume
that t, s: V → V are each diagonalizable. Is ct diagonalizable for all scalars c?
What about t + s? t ◦ s?
2.19 Show that matrices of this form are not diagonalizable.
( ) 1 c
c ≠ 0
0 1
2.20 Show
(
that
)
each of
(
these
)
is diagonalizable.
1 2 x y
(a)
(b)
x, y, z scalars
2 1 y z
II.3
Eigenvalues and Eigenvectors
We will next focus on the property of Lemma 2.4.
3.1 Definition A transformation t: V → V has a scalar eigenvalue λ if there is a
nonzero eigenvector ⃗ζ ∈ V such that t(⃗ζ) =λ · ⃗ζ.
“Eigen” is German for “characteristic of” or “peculiar to.” Some authors call these
characteristic values and vectors. No authors call them “peculiar” vectors.
3.2 Example The projection map
⎛ ⎞ ⎛ ⎞
x
⎜ ⎟
⎝y⎠
z
π
↦−→
x
⎜ ⎟
⎝y⎠
0
x, y, z ∈ C
has an eigenvalue of 1 associated with any eigenvector
⎛ ⎞
x
⎜ ⎟
⎝y⎠
0