- Exercises
Compute the
characteristic polynomial
of the
matrix
-
Compute the
characteristic polynomial,
the
eigenvalues
and the
eigenspaces
of the matrix
-
over
.
Show that the characteristic polynomial of a
linear mapping
on a
finite-dimensional
-vector space
is well-defined, that is, independent of the chosen
basis.
Let
be a
field
and let
denote an
-matrix
over
. Show that for every
,
the relation
-
![{\displaystyle {}\chi _{M}(\lambda )=\det {\left(\lambda E_{n}-M\right)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8fb803fa1e1dbf6ffdce96e75a32eed0e9f25a9a)
holds.[1]
Let
be a field and let
be an
-matrix
over
. Where can you find the
determinant
of
within the
characteristic polynomial
?
Show that the
characteristic polynomial
of the so-called Companion matrix
-
![{\displaystyle {}M={\begin{pmatrix}0&1&0&\ldots &0\\0&0&1&\ldots &0\\\vdots &\vdots &\ddots &\ddots &\vdots \\0&0&\ldots &0&1\\-a_{0}&-a_{1}&\ldots &-a_{n-2}&-a_{n-1}\end{pmatrix}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e91fcec2aa95f3ed1dfe545d9497383ac24181c)
equals
-
![{\displaystyle {}\chi _{M}=X^{n}+a_{n-1}X^{n-1}+\cdots +a_{1}X+a_{0}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e936335548d5bcf91b451b0ae2252735660300b)
We consider the real matrix
-
![{\displaystyle {}M={\begin{pmatrix}1&1\\1&0\end{pmatrix}}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0954694330640b54bf62189c40586199895b71b6)
a) Determine
-
for
.
b) Let
-
![{\displaystyle {}{\begin{pmatrix}x_{n+1}\\y_{n+1}\end{pmatrix}}:=M^{n}{\begin{pmatrix}1\\0\end{pmatrix}}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac4b6aa2617424cb16e4f285ce089b143da20816)
Establish a relation between the sequences
and
, and determine a recursive formula for these sequences.
c) Determine the eigenvalues and the eigenvectors of
.
Let
-
![{\displaystyle {}M={\begin{pmatrix}-1&1&0&0\\0&-1&1&0\\0&0&-1&1\\-1&0&0&1\end{pmatrix}}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/900f86b24cd610d3f5149551eddec64ad8c51f17)
- Determine the
characteristic polynomial
of
.
- Determine a zero of the characteristic polynomial of
, and write the polynomial using the corresponding linear factor.
- Show that the characteristic polynomial of
has at least two real roots.
Let
be a zero of the polynomial
-
Show that
-
is an
eigenvector
of the matrix
-
for the
eigenvalue
.
To solve the following exercise, the two exercises above and also
Exercise 24.31
are helpful.
We consider the mapping
-
which assigns to a four tuple
the four tuple
-
Show that there exists a tuple
, for which arbitrary iterations of the mapping do never reach the zero tupel.
Determine the eigenvalues and the eigenspaces of the linear mapping
-
given by the matrix
-
![{\displaystyle {}M={\begin{pmatrix}2&0&5\\0&-1&0\\8&0&5\end{pmatrix}}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/846356e65b4eafea649790d5fb9a01357bb912a8)
We consider the linear mapping
-
which is given by the matrix
-
![{\displaystyle {}A={\begin{pmatrix}2&1&-2+{\mathrm {i} }\\0&{\mathrm {i} }&1+{\mathrm {i} }\\0&0&-1+2{\mathrm {i} }\end{pmatrix}}\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6796f8c6b883d146ccc6f0a42475edb816fc01d)
with respect to the standard basis.
a) Determine the characteristic polynomial and the eigenvalues of
.
b) Compute, for every eigenvalue, an eigenvector.
c) Establish a matrix for
with respect to a basis of eigenvectors.
Let
-
![{\displaystyle {}A={\begin{pmatrix}-4&6&6\\0&2&0\\-3&3&5\end{pmatrix}}\in \operatorname {Mat} _{3\times 3}(\mathbb {R} )\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e56fad511aedba179f2936b5ab377645b31e76c0)
Compute:
- the eigenvalues of
;
- the corresponding eigenspaces;
- the geometric and algebraic multiplicities of each eigenvalue;
- a matrix
such that
is a diagonal matrix.
Determine the
eigenspace
and the
geometric multiplicity
for
of the matrix
-
Show that the
matrix
-
is
diagonalizable
over
.
Let
be a
matrix
with
(pairwise)
different
eigenvalues.
Show that the
determinant
of
is the product of the eigenvalues.
Let
be a field,
and
numbers with
.
Give an example of an
-matrix
, such that
is an
eigenvalue
for
with
algebraic multiplicity
and
geometric multiplicity
.
Determine, which of the following elementary-geometric mappings are linear, which are diagonalizable and which are trigonalizable.
- The reflection in the plane, given by the line
as axis.
- The translation with the vector
.
- The rotation by
degree counter-clockwise around the origin.
- The reflection with
as center.
Determine, whether the real matrix
-
is
trigonalizable
or not.
Suppose that a
linear mapping
-
is given by the
matrix
-
with respect to the
standard basis.
Find a
basis,
such that
is described by the matrix
-
with respect to this basis.
The next exercises use the following definition.
Let
a
linear mapping
on a
-vector space
over a field
. Show the following properties.
- The
zero space
is
-invariant.
is
-invariant.
- Eigenspaces
are
-invariant.
- Let
be
-invariant linear subspaces. Then also
and
are
-invariant.
- Let
be a
-invariant linear subspace. Then also the
image space
and the
preimage space
are
-invariant.
Let
a
linear mapping
on a
-vector space
over a field
, and let
.
Show that the smallest
-invariant linear subspace
of
, which contains
, equals
-
Let
a
linear mapping
on a
-vector space
over a field
. Show that the subset of
, defined by
-
![{\displaystyle {}U={\left\{v\in V\mid {\text{ there exists an }}n\in \mathbb {N} {\text{ with }}\varphi ^{n}(v)=0\right\}}\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ef04e0a8abf7655ae17a684d3a2346b99536a03)
is an
-invariant linear subspace.
Let
be a
linear mapping
on a
-vector space
. Let
be a
basis
of
, such that
is described, with respect to this basis, by an
upper triangular matrix.
Show that the
linear subspaces
-
are
-invariant
for every
.
Determine, whether the real matrix
-
is
trigonalizable
or not.
- Hand-in-exercises
Compute the
characteristic polynomial
of the
matrix
-
Compute the
characteristic polynomial,
the
eigenvalues
and the
eigenspaces
of the matrix
-
over
.
Let
-
be a
linear mapping.
Show that
has at least one
eigenvector.
Let
-
![{\displaystyle {}A={\begin{pmatrix}-5&0&7\\6&2&-6\\-4&0&6\end{pmatrix}}\in \operatorname {Mat} _{3\times 3}(\mathbb {R} )\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e87171fa2307d59f4a4e087316f5de63211fa358)
Compute:
- the eigenvalues of
;
- the corresponding eigenspaces;
- the geometric and algebraic multiplicities of each eigenvalue;
- a matrix
such that
is a diagonal matrix.
Determine for every
the
algebraic
and
geometric
multiplicities for the
matrix
-
![{\displaystyle {}M={\begin{pmatrix}3&-4&5\\0&-1&2\\0&0&3\end{pmatrix}}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78dcd503f09bec87ccce6ed8fd84ecaedc1db8bf)
Decide whether the
matrix
-
is
trigonalizable
over
.
Determine whether the real matrix
-
is
trigonalizable
or not.
- Footnotes
- ↑ The main difficulty might be here to recognize that there is indeed something to show.