Exercises Determine the
eigenvectors
of the function
R
→
R
,
x
↦
π
x
{\displaystyle {}\mathbb {R} \rightarrow \mathbb {R} ,x\mapsto \pi x}
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 .
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}
Determine the eigenvalues, eigenvectors and eigenspaces for a
plane rotation
(
cos
α
−
sin
α
sin
α
cos
α
)
{\displaystyle {}{\begin{pmatrix}\operatorname {cos} \,\alpha &-\operatorname {sin} \,\alpha \\\operatorname {sin} \,\alpha &\operatorname {cos} \,\alpha \end{pmatrix}}}
α
{\displaystyle {}\alpha }
0
≤
α
<
2
π
{\displaystyle {}0\leq \alpha <2\pi }
R
{\displaystyle {}\mathbb {R} }
Show that every
matrix
M
∈
Mat
2
(
C
)
{\displaystyle {}M\in \operatorname {Mat} _{2}(\mathbb {C} )}
eigenvalue .
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 }
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}}
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}
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} }
[ 1] eigenvalue
λ
{\displaystyle {}\lambda }
φ
{\displaystyle {}\varphi }
λ
n
=
1
{\displaystyle {}\lambda ^{n}=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 )}
[ 2]
P
(
φ
)
{\displaystyle {}P(\varphi )}
Let
M
{\displaystyle {}M}
M
=
(
A
0
0
B
)
{\displaystyle {}M={\begin{pmatrix}A&0\\0&B\end{pmatrix}}\,}
with square matrices
A
{\displaystyle {}A}
B
{\displaystyle {}B}
λ
∈
K
{\displaystyle {}\lambda \in K}
eigenvalue
of
M
{\displaystyle {}M}
λ
{\displaystyle {}\lambda }
A
{\displaystyle {}A}
B
{\displaystyle {}B}
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 the following statements hold.
Every
eigenspace
Eig
λ
(
φ
)
{\displaystyle \operatorname {Eig} _{\lambda }{\left(\varphi \right)}}
is a
linear subspace
of
V
{\displaystyle {}V}
λ
{\displaystyle {}\lambda }
eigenvalue
for
φ
{\displaystyle {}\varphi }
Eig
λ
(
φ
)
{\displaystyle {}\operatorname {Eig} _{\lambda }{\left(\varphi \right)}}
A vector
v
∈
V
,
v
≠
0
{\displaystyle {}v\in V,\,v\neq 0}
eigenvector
for
λ
{\displaystyle {}\lambda }
v
∈
Eig
λ
(
φ
)
{\displaystyle {}v\in \operatorname {Eig} _{\lambda }{\left(\varphi \right)}}
Let
V
=
R
[
X
]
≤
d
{\displaystyle {}V=\mathbb {R} [X]_{\leq d}}
≤
d
{\displaystyle {}\leq d}
V
⟶
V
,
P
⟼
P
′
.
{\displaystyle V\longrightarrow V,P\longmapsto P'.}
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 .[ 3]
eigenspace
and its
dimension .
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
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 .
Let
λ
1
≠
λ
2
{\displaystyle {}\lambda _{1}\neq \lambda _{2}}
K
{\displaystyle {}K}
Eig
λ
1
(
φ
)
∩
Eig
λ
2
(
φ
)
=
0
.
{\displaystyle {}\operatorname {Eig} _{\lambda _{1}}{\left(\varphi \right)}\cap \operatorname {Eig} _{\lambda _{2}}{\left(\varphi \right)}=0\,.}
Let
K
{\displaystyle {}K}
field ,
V
{\displaystyle {}V}
finite-dimensional
K
{\displaystyle {}K}
vector space
and
φ
:
V
⟶
V
{\displaystyle \varphi \colon V\longrightarrow V}
a
linear mapping .
Show that there exist at most
dim
K
(
V
)
{\displaystyle {}\dim _{K}{\left(V\right)}}
eigenvalues
for
φ
{\displaystyle {}\varphi }
Hand-in-exercises Check whether the vector
(
6
−
5
8
)
{\displaystyle {}{\begin{pmatrix}6\\-5\\8\end{pmatrix}}}
eigenvector
of the matrix
(
−
8
0
−
1
2
4
−
5
−
3
7
2
)
.
{\displaystyle {\begin{pmatrix}-8&0&-1\\2&4&-5\\-3&7&2\end{pmatrix}}.}
In this case, determine the corresponding
eigenvalue .
Check whether the vector
(
7
2
3
)
{\displaystyle {}{\begin{pmatrix}7\\2\\3\end{pmatrix}}}
eigenvector
of the matrix
(
2
−
4
−
9
−
2
7
−
2
−
1
−
1
0
)
.
{\displaystyle {\begin{pmatrix}2&-4&-9\\-2&7&-2\\-1&-1&0\end{pmatrix}}.}
In this case, determine the corresponding
eigenvalue .
The nightlife in the village Kleineisenstein consists in the following three opportunities: to stay in bed
(at home),
the pub "Nightowl“ and the dancing club "Pirouette“. In a night, one can observe within an hour the following movements:
a)
1
/
10
{\displaystyle {}1/10}
1
/
12
{\displaystyle {}1/12}
1
/
3
{\displaystyle {}1/3}
1
/
5
{\displaystyle {}1/5}
3
/
5
{\displaystyle {}3/5}
8
{\displaystyle {}8}
Establish a matrix, which describes the movements within an hour.
Kleineisenstein has
500
{\displaystyle {}500}
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.
Footnotes
↑ The value
n
=
0
{\displaystyle {}n=0}
↑ The expression
P
(
φ
)
{\displaystyle P(\varphi )}
φ
{\displaystyle {}\varphi }
P
{\displaystyle {}P}
X
n
{\displaystyle {}X^{n}}
φ
n
{\displaystyle {}\varphi ^{n}}
n
{\displaystyle {}n}
φ
{\displaystyle {}\varphi }
↑ In this context, one also says eigenfunction instead of eigenvector.
![{\displaystyle {}P\in K[X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4692e3ffcbd66e900f8975113ff3f32f1cabc21a)
![{\displaystyle {}V=\mathbb {R} [X]_{\leq d}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16f20283b7a42ff12c2a38ee1cd8d81b3a98238c)
