Exercise for the break Check whether the vector
(
3
1
−
1
)
{\displaystyle {}{\begin{pmatrix}3\\1\\-1\end{pmatrix}}}
eigenvector
for the matrix
(
−
2
−
5
1
0
−
2
2
4
−
3
5
)
.
{\displaystyle {\begin{pmatrix}-2&-5&1\\0&-2&2\\4&-3&5\end{pmatrix}}.}
In this case, determine the corresponding
eigenvalue .
Exercises Let a
linear mapping
φ
:
K
⟶
K
{\displaystyle \varphi \colon K\longrightarrow K}
be given. What are the
eigenvalues
and the
eigenvectors
of
φ
{\displaystyle {}\varphi }
Let
φ
,
ψ
:
V
⟶
V
{\displaystyle \varphi ,\psi \colon V\longrightarrow V}
be
endomorphisms
on a
K
{\displaystyle {}K}
vector space
V
{\displaystyle {}V}
v
∈
V
{\displaystyle {}v\in V}
eigenvector
of
φ
{\displaystyle {}\varphi }
ψ
{\displaystyle {}\psi }
v
{\displaystyle {}v}
φ
∘
ψ
{\displaystyle {}\varphi \circ \psi }
Determine the
eigenvectors
and the
eigenvalues
for a
linear mapping
φ
:
R
2
⟶
R
2
,
{\displaystyle \varphi \colon \mathbb {R} ^{2}\longrightarrow \mathbb {R} ^{2},}
given by a matrix of the form
(
a
b
0
d
)
{\displaystyle {}{\begin{pmatrix}a&b\\0&d\end{pmatrix}}}
Show that the first
standard vector
is an eigenvector for every
upper triangular matrix .
What is its
eigenvalue ?
Let
M
=
(
d
1
∗
⋯
⋯
∗
0
d
2
∗
⋯
∗
⋮
⋱
⋱
⋱
⋮
0
⋯
0
d
n
−
1
∗
0
⋯
⋯
0
d
n
)
{\displaystyle {}M={\begin{pmatrix}d_{1}&\ast &\cdots &\cdots &\ast \\0&d_{2}&\ast &\cdots &\ast \\\vdots &\ddots &\ddots &\ddots &\vdots \\0&\cdots &0&d_{n-1}&\ast \\0&\cdots &\cdots &0&d_{n}\end{pmatrix}}\,}
be an
upper triangular matrix .
Show that an
eigenvalue
of
M
{\displaystyle {}M}
M
{\displaystyle {}M}
Give an example of a
linear mapping
φ
:
R
2
⟶
R
2
{\displaystyle \varphi \colon \mathbb {R} ^{2}\longrightarrow \mathbb {R} ^{2}}
such that
φ
{\displaystyle {}\varphi }
eigenvalue ,
but a certain
power
φ
n
{\displaystyle {}\varphi ^{n}}
n
≥
2
{\displaystyle {}n\geq 2}
Show that every
matrix
M
∈
Mat
2
(
C
)
{\displaystyle {}M\in \operatorname {Mat} _{2}(\mathbb {C} )}
eigenvalue .
Let
K
{\displaystyle {}K}
field ,
V
{\displaystyle {}V}
K
{\displaystyle {}K}
vector space
and
φ
:
V
⟶
V
{\displaystyle \varphi \colon V\longrightarrow V}
a
linear mapping .
Show that
ker
(
φ
)
=
Eig
0
(
φ
)
.
{\displaystyle {}\operatorname {ker} {\left(\varphi \right)}=\operatorname {Eig} _{0}{\left(\varphi \right)}\,.}
Let
K
{\displaystyle {}K}
φ
:
V
→
V
{\displaystyle {}\varphi \colon V\rightarrow V}
endomorphism
on a
K
{\displaystyle {}K}
vector space
V
{\displaystyle {}V}
λ
∈
K
{\displaystyle {}\lambda \in K}
U
=
Eig
λ
(
φ
)
{\displaystyle {}U=\operatorname {Eig} _{\lambda }{\left(\varphi \right)}\,}
be the corresponding
eigenspace .
Show that
φ
{\displaystyle {}\varphi }
φ
|
U
:
U
⟶
U
,
v
⟼
φ
(
v
)
,
{\displaystyle \varphi {|}_{U}\colon U\longrightarrow U,v\longmapsto \varphi (v),}
and that this mapping is the
homothety
with scale factor
λ
{\displaystyle {}\lambda }
Let
φ
:
V
→
V
{\displaystyle {}\varphi \colon V\rightarrow V}
isomorphism
on a
K
{\displaystyle {}K}
vector space
V
{\displaystyle {}V}
φ
−
1
{\displaystyle {}\varphi ^{-1}}
inverse mapping .
Show that
a
∈
K
{\displaystyle {}a\in K}
eigenvalue
of
φ
{\displaystyle {}\varphi }
a
−
1
{\displaystyle {}a^{-1}}
φ
−
1
{\displaystyle {}\varphi ^{-1}}
Let
K
{\displaystyle {}K}
φ
:
V
→
V
{\displaystyle {}\varphi \colon V\rightarrow V}
endomorphism
on a
K
{\displaystyle {}K}
vector space
V
{\displaystyle {}V}
λ
∈
K
{\displaystyle {}\lambda \in K}
eigenvalue
of
φ
{\displaystyle {}\varphi }
P
∈
K
[
X
]
{\displaystyle {}P\in K[X]}
polynomial .
Show that
P
(
λ
)
{\displaystyle {}P(\lambda )}
P
(
φ
)
{\displaystyle {}P(\varphi )}
Let
V
1
,
…
,
V
n
{\displaystyle {}V_{1},\ldots ,V_{n}}
vector spaces
over a
field
K
{\displaystyle {}K}
φ
i
:
V
i
⟶
V
i
{\displaystyle \varphi _{i}\colon V_{i}\longrightarrow V_{i}}
denote
linear mappings .
Let
a
∈
K
{\displaystyle {}a\in K}
eigenvalue
of
φ
k
{\displaystyle {}\varphi _{k}}
k
{\displaystyle {}k}
a
{\displaystyle {}a}
φ
1
×
⋯
×
φ
n
:
V
1
×
⋯
×
V
n
⟶
V
1
×
⋯
×
V
n
.
{\displaystyle \varphi _{1}\times \cdots \times \varphi _{n}\colon V_{1}\times \cdots \times V_{n}\longrightarrow V_{1}\times \cdots \times V_{n}.}
Show that
λ
∈
K
{\displaystyle {}\lambda \in K}
eigenvalue
for the
linear mapping
φ
:
K
2
⟶
K
2
{\displaystyle \varphi \colon K^{2}\longrightarrow K^{2}}
given by a matrix of the form
(
a
b
c
d
)
{\displaystyle {}{\begin{pmatrix}a&b\\c&d\end{pmatrix}}}
λ
{\displaystyle {}\lambda }
X
2
−
(
a
+
d
)
X
+
a
d
−
c
b
.
{\displaystyle X^{2}-(a+d)X+ad-cb.}
Let
V
{\displaystyle {}V}
R
{\displaystyle {}\mathbb {R} }
R
{\displaystyle {}\mathbb {R} }
a) Show that the derivation
f
↦
f
′
{\displaystyle {}f\mapsto f'}
linear mapping
from
V
{\displaystyle {}V}
V
{\displaystyle {}V}
eigenvalues
of the derivation and determine, for each eigenvalue, at least one
eigenvector .[ 1]
eigenspace
and its
dimension .
Let
φ
:
V
⟶
V
{\displaystyle \varphi \colon V\longrightarrow V}
be an
endomorphism
on a
finite-dimensional
K
{\displaystyle {}K}
vector space
V
{\displaystyle {}V}
v
∈
V
{\displaystyle {}v\in V}
eigenvector
for
φ
{\displaystyle {}\varphi }
eigenvalue
λ
∈
K
{\displaystyle {}\lambda \in K}
φ
∗
:
V
∗
⟶
V
∗
{\displaystyle {\varphi }^{*}\colon {V}^{*}\longrightarrow {V}^{*}}
be the
dual mapping
of
φ
{\displaystyle {}\varphi }
V
{\displaystyle {}V}
v
,
u
1
,
…
,
u
r
{\displaystyle {}v,u_{1},\ldots ,u_{r}}
v
∗
,
u
1
∗
,
…
,
u
r
∗
{\displaystyle {}v^{*},u_{1}^{*},\ldots ,u_{r}^{*}}
a)
v
∗
{\displaystyle {}v^{*}}
φ
∗
{\displaystyle {}{\varphi }^{*}}
λ
{\displaystyle {}\lambda }
u
1
,
…
,
u
r
{\displaystyle {}u_{1},\ldots ,u_{r}}
v
∗
{\displaystyle {}v^{*}}
φ
∗
{\displaystyle {}{\varphi }^{*}}
λ
{\displaystyle {}\lambda }
v
,
u
1
,
…
,
u
r
{\displaystyle {}v,u_{1},\ldots ,u_{r}}
v
,
w
1
,
…
,
w
r
{\displaystyle {}v,w_{1},\ldots ,w_{r}}
v
∗
{\displaystyle {}v^{*}}
v
,
u
1
,
…
,
u
r
{\displaystyle {}v,u_{1},\ldots ,u_{r}}
φ
∗
{\displaystyle {}{\varphi }^{*}}
Let
V
{\displaystyle {}V}
finite-dimensional
K
{\displaystyle {}K}
vector space ,
and
v
∈
V
{\displaystyle {}v\in V}
v
≠
0
{\displaystyle {}v\neq 0}
R
(
v
)
=
{
φ
∈
End
(
V
)
∣
v
is eigenvector of
φ
}
,
{\displaystyle {}R(v)={\left\{\varphi \in \operatorname {End} _{}{\left(V\right)}\mid v{\text{ is eigenvector of }}\varphi \right\}}\,,}
with the natural addition and multiplication of endomorphisms, is a
ring
and a
linear subspace
of
End
(
V
)
{\displaystyle {}\operatorname {End} _{}{\left(V\right)}}
dimension
of this space.
Let
K
{\displaystyle {}K}
field ,
c
∈
K
{\displaystyle {}c\in K}
a
=
(
a
1
⋮
a
n
)
∈
K
n
{\displaystyle {}a={\begin{pmatrix}a_{1}\\\vdots \\a_{n}\end{pmatrix}}\in K^{n}}
0
{\displaystyle {}0}
inhomogeneous system of linear equations
such that its
solution set
is the set of all
n
×
n
{\displaystyle {}n\times n}
matrices ,
for that
a
{\displaystyle {}a}
eigenvector
with
eigenvalue
c
{\displaystyle {}c}
Let
M
{\displaystyle {}M}
n
×
n
{\displaystyle {}n\times n}
matrix .
Let
a
∈
R
{\displaystyle {}a\in \mathbb {R} }
eigenvalue
of
M
{\displaystyle {}M}
a
{\displaystyle {}a}
R
{\displaystyle {}\mathbb {R} }
M
{\displaystyle {}M}
Generalize the previous statement for any
field extension
K
⊆
L
{\displaystyle {}K\subseteq L}
Hand-in-exercises Let
K
{\displaystyle {}K}
φ
:
V
→
V
{\displaystyle {}\varphi \colon V\rightarrow V}
endomorphism
on a
K
{\displaystyle {}K}
vector space
V
{\displaystyle {}V}
φ
{\displaystyle {}\varphi }
homothety
if and only if every vector
v
∈
V
{\displaystyle {}v\in V}
v
≠
0
{\displaystyle {}v\neq 0}
eigenvector
of
φ
{\displaystyle {}\varphi }
Consider the matrix
M
=
(
1
1
−
1
1
)
.
{\displaystyle {}M={\begin{pmatrix}1&1\\-1&1\end{pmatrix}}\,.}
Show that
M
{\displaystyle {}M}
eigenvalue .
Determine the eigenvalues and the
eigenspaces
of
M
{\displaystyle {}M}
complex
matrix.
Consider the
real
matrices
(
a
b
c
d
)
∈
Mat
2
(
R
)
.
{\displaystyle {}{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in \operatorname {Mat} _{2}(\mathbb {R} )\,.}
Characterize, in dependence on
a
,
b
,
c
,
d
{\displaystyle {}a,b,c,d}
two different
eigenvalues ,
one eigenvalue with a two-dimensional
eigenspace ,
one eigenvalue with a one-dimensional
eigenspace ,
no eigenvalue.
Let
K
{\displaystyle {}K}
φ
:
V
→
V
{\displaystyle {}\varphi \colon V\rightarrow V}
endomorphism
on a
K
{\displaystyle {}K}
vector space
V
{\displaystyle {}V}
φ
n
=
Id
V
{\displaystyle {}\varphi ^{n}=\operatorname {Id} _{V}\,}
for a certain
n
∈
N
{\displaystyle {}n\in \mathbb {N} }
[ 2] eigenvalue
λ
{\displaystyle {}\lambda }
φ
{\displaystyle {}\varphi }
λ
n
=
1
{\displaystyle {}\lambda ^{n}=1}
Let
K
{\displaystyle {}K}
field ,
and let
V
{\displaystyle {}V}
n
{\displaystyle {}n}
dimensional
K
{\displaystyle {}K}
vector space .
Let
φ
:
V
⟶
V
{\displaystyle \varphi \colon V\longrightarrow V}
be a
linear mapping .
Let
λ
≠
0
{\displaystyle {}\lambda \neq 0}
eigenvalue
of
φ
{\displaystyle {}\varphi }
v
{\displaystyle {}v}
eigenvector .
Show that, for a given
basis
v
,
u
2
,
…
,
u
n
{\displaystyle {}v,u_{2},\ldots ,u_{n}}
V
{\displaystyle {}V}
v
,
w
2
,
…
,
w
n
{\displaystyle {}v,w_{2},\ldots ,w_{n}}
⟨
v
,
u
j
⟩
=
⟨
v
,
w
j
⟩
{\displaystyle {}\langle v,u_{j}\rangle =\langle v,w_{j}\rangle }
φ
(
w
j
)
∈
⟨
u
i
,
i
=
2
,
…
,
n
⟩
{\displaystyle {}\varphi (w_{j})\in \langle u_{i},\,i=2,\ldots ,n\rangle \,}
for all
j
=
2
,
…
,
n
{\displaystyle {}j=2,\ldots ,n}
Show also that this is not possible for
λ
=
0
{\displaystyle {}\lambda =0}
Footnotes
↑ In this context, one also says eigenfunction instead of eigenvector.
↑ The value
n
=
0
{\displaystyle {}n=0}
![{\displaystyle {}P\in K[X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4692e3ffcbd66e900f8975113ff3f32f1cabc21a)
