- Exercise for the break
Compute the
characteristic polynomial
of the
matrix
-
- Exercises
Determine the characteristic polynomial and the eigenvalues of the linear mapping
-
given by the matrix
-
with respect to the standard 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
?
Let be a
field,
and let denote an
-matrix
over . How can we find the in the
characteristic polynomial
?
Determine the
characteristic polynomial
of a matrix
-
What is the relevance of the coefficients of this polynomial?
Determine the
characteristic polynomial
of a matrix
-
What is the relevance of the coefficients of this polynomial?
Compute the
characteristic polynomial
of the matrix
-
over the
field of rational functions
.
Compute the
characteristic polynomial,
the
eigenvalues
and the
eigenspaces
of the matrix
-
over .
Determine the
characteristic polynomial,
the
eigenvalues,
and the
eigenspaces
of the
matrix
-
over .
Determine the eigenvalues and the eigenspaces of the linear mapping
-
given by the matrix
-
We consider the linear mapping
-
that 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.
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, besides the preceding exercises also
Exercise 10.16
is helpful.
We consider the mapping
-
that assigns to a four tuple the four tuple
-
Show that there exists a tuple , for that arbitrary iterations of the mapping do never reach the zero tuple.
Let be a
field,
and let denote an
-matrix
over with the property, that the
characteristic polynomial
splits into linear factors, that is,
-
Show that
-
Let be the
field with two elements,
we consider the
matrix
-
over . Show that the
characteristic polynomial
is not the zero polynomial, but that
-
holds for all
.
Show that a square matrix and its
transposed matrix
have the same
characteristic polynomial.
What is wrong in the following argumentation:
"For two
-matrices
, the
characteristic polynomials
fulfill the relation
-
This is because, by definition, we have
-
where the equation in the middle rests on the multiplication theorem for determinants“.
Let be an
-matrix,
with the
characteristic polynomial
-
Determine the characteristic polynomial of the scaled matrix ,
.
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
.
Let
be a
field extension.
Let an
-matrix
over be given. Show that the
characteristic polynomial
coincides with the characteristic polynomial of , considered as a 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
finite-dimensional
-vector space, and let
.
Show that the following statements are equivalent:
- The linear mapping is an isomorphism.
- is not an eigenvalue of .
- The constant term of the characteristic polynomial is .
Let
-
be an
endomorphism
on a
finite-dimensional
-vector space
, and let
an
eigenvalue
of . Show that is also an eigenvalue of the
dual mapping
-
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 denote a
field,
and let denote a
-vector space
of finite dimension. Let
-
be a
linear mapping.
Suppose that the
characteristic polynomial
factors into different
linear factors.
Show that is
diagonalizable.
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 that contains , equals
-
Let be a
field,
let be a
-vector space,
and let
-
a
linear mapping. Let
be a
-invariant linear subspace
of . Show that, for a polynomial
,
the space is also -invariant.
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 .
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
finite-dimensional
-vector space
. Let
.
Show that there exists an
invariant linear subspace
of dimension , if and only if there exists a
basis
of such that the
describing matrix
of , with respect to this basis, has the form
-
Let
be a
linear mapping
on the
finite-dimensional
-vector space
. Let
.
Show that there exists a
direct sum decomposition
into
invariant linear subspaces
of dimension and , if and only if there exists a
basis
of such that the
describing matrix
of with respect to this basis has the form
-
Let be a
finite-dimensional
-vector space,
and
a
linear subspace.
Show that
-
is, with the natural addition and multiplication of endomorphisms, a
ring,
and a
linear subspace
of . Determine the
dimension
of this space.
- Hand-in-exercises
Compute the
characteristic polynomial
of the
matrix
-
Compute the
characteristic polynomial,
the
eigenvalues
and the
eigenspaces
of the matrix
-
over .
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
-
Show that the
characteristic polynomial
of the so-called companion matrix
-
equals
-
Let
-
be a
linear mapping.
Show that has at least one
eigenvector.
- Footnotes
- ↑ The main difficulty might be here to recognize that there is indeed something to show.