- 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
-
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
-
equals
-
We consider the real matrix
-
a) Determine
-
for
.
b) Let
-
Establish a relation between the sequences and , and determine a recursive formula for these sequences.
c) Determine the eigenvalues and the eigenvectors of .
Let
-
- 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
-
We consider the linear mapping
-
which is given by the matrix
-
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
-
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
-
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
-
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
-
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.