# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 27

*Exercises*

Determine the eigenvectors of the function .

Check whether the vector is an eigenvector for the matrix

In this case, determine the corresponding eigenvalue.

Determine the eigenvectors and the eigenvalues for a linear mapping

given by a matrix of the form .

Show that the first standard vector is an eigenvector for every upper triangular matrix. What is its eigenvalue?

Let

be an upper triangular matrix. Show that an eigenvalue of is a diagonal entry of .

Determine the eigenvalues, eigenvectors and eigenspaces for a plane rotation , with rotation angle , , over .

Show that every matrix has at least one eigenvalue.

Let

be endomorphisms on a -vector space , and let be an eigenvector of and of . Show that is also an eigenvector of . What is its eigenvalue?

Let be an isomorphism on a -vector space , and let be its inverse mapping. Show that is an eigenvalue of if and only if is an eigenvalue of .

Give an example of a linear mapping

such that has no eigenvalue, but a certain power , , has an eigenvalue.

Let be a field, and let be an endomorphism on a -vector space , satisfying

for a certain
.^{[1]}
Show that every
eigenvalue
of fulfills the property
.

Let be a field, and let
be an
endomorphism
on a
-vector space
. Let
be an
eigenvalue
of and
a
polynomial.
Show that is an eigenvalue^{[2]} of .

Let be a square matrix, which can be written as a block matrix

with square matrices and . Show that a number is an eigenvalue of if and only if is an eigenvalue of or of .

Let be a field, a -vector space and

a linear mapping. Show that the following statements hold.

- Every
eigenspace
is a linear subspace of .

- is an eigenvalue for if and only if the eigenspace is not the null space.
- A vector , is an eigenvector for if and only if .

Let denote the set of all real polynomials of degree . Determine the eigenvalues, eigenvectors and eigenspaces of the derivation operator

The concept of an eigenvector is also defined for vector spaces of infinite dimension, the relevance can be seen already in the following exercise.

Let denote the real vector space that consists of all functions from to that are arbitrarily often differentiable.

a) Show that the derivation is a linear mapping from to .

b) Determine the
eigenvalues
of the derivation and determine, for each eigenvalue, at least one
eigenvector.^{[3]}

c) Determine for every real number the
eigenspace
and its
dimension.

Let be a field, a -vector space and

a linear mapping. Show that

Let be a field, and let be an endomorphism on a -vector space . Let and let

be the corresponding eigenspace. Show that can be restricted to a linear mapping

and that this mapping is the homothety with scale factor .

Let be a field, a -vector space and

a linear mapping. Let be elements in . Show that

Let be a field, a finite-dimensional -vector space and

a linear mapping. Show that there exist at most many eigenvalues for .

*Hand-in-exercises*

### Exercise (1 mark)

Check whether the vector is an eigenvector of the matrix

In this case, determine the corresponding eigenvalue.

### Exercise (1 mark)

Check whether the vector is an eigenvector of the matrix

In this case, determine the corresponding eigenvalue.

### Exercise (4 (1+3) marks)

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) of the people in bed go to the Nightowl, go to the Pirouette and the rest stays in bed.

b) of the people in the Nightowl go to the Pirouette, go to bed and the rest stays in the Nightowl.

c) of the people in the Pirouette stay in the Pirouette, a percentage of go to the Nightowl, the rest goes to bed.

- Establish a matrix, which describes the movements within an hour.
- Kleineisenstein has inhabitants. Determine a distribution of the inhabitants (on the three locations), which does not change within an hour.

### Exercise (3 marks)

Let be a field, and let be an endomorphism on a -vector space . Show that is a homothety if and only if every vector , , is an eigenvector of .

### Exercise (4 marks)

Consider the matrix

Show that , as a real matrix, has no eigenvalue. Determine the eigenvalues and the eigenspaces of as a complex matrix.

### Exercise (6 marks)

Consider the real matrices

Characterize, in dependence on , when such a matrix has

- two different eigenvalues,
- one eigenvalue with a two-dimensional eigenspace,
- one eigenvalue with a one-dimensional eigenspace,
- no eigenvalue.

*Footnotes*

- ↑ The value is allowed, but does not say much.
- ↑ The expression means that the linear mapping is inserted into the polynomial . Here, has to be interpreted as , the -th composition of with itself. The addition becomes the addition of linear mappings, etc.
- ↑ In this context, one also says
*eigenfunction*instead of eigenvector.

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