# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 28

*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 be a field, a vector space over and

a
linear mapping.
A
linear subspace
is called
-*invariant*,
if

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*

### Exercise (2 marks)

Compute the characteristic polynomial of the matrix

### Exercise (3 marks)

Compute the characteristic polynomial, the eigenvalues and the eigenspaces of the matrix

over .

### Exercise (4 marks)

Let

be a linear mapping. Show that has at least one eigenvector.

### Exercise (4 marks)

Let

Compute:

- the eigenvalues of ;
- the corresponding eigenspaces;
- the geometric and algebraic multiplicities of each eigenvalue;
- a matrix such that is a diagonal matrix.

### Exercise (4 marks)

Determine for every the algebraic and geometric multiplicities for the matrix

### Exercise (4 marks)

Decide whether the matrix

is trigonalizable over .

### Exercise (3 marks)

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.

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