Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 27/refcontrol
- Exercises
Exercise Create referencenumber
Determine the eigenvectorsMDLD/eigenvectors of the function .
Exercise Create referencenumber
Check whether the vector is an eigenvectorMDLD/eigenvector for the matrix
In this case, determine the corresponding eigenvalue.MDLD/eigenvalue
Exercise Create referencenumber
Determine the eigenvectorsMDLD/eigenvectors and the eigenvaluesMDLD/eigenvalues for a linear mappingMDLD/linear mapping
given by a matrix of the form .
Exercise Create referencenumber
Show that the first standard vectorMDLD/standard vector is an eigenvector for every upper triangular matrix.MDLD/upper triangular matrix What is its eigenvalue?MDLD/eigenvalue
Exercise Create referencenumber
Let
be an upper triangular matrix.MDLD/upper triangular matrix Show that an eigenvalueMDLD/eigenvalue of is a diagonal entry of .
Exercise Create referencenumber
Determine the eigenvalues, eigenvectors and eigenspaces for a plane rotationMDLD/plane rotation , with rotation angle , , over .
Exercise Create referencenumber
Show that every matrixMDLD/matrix has at least one eigenvalue.MDLD/eigenvalue
Exercise Create referencenumber
Let
be an endomorphismsMDLD/endomorphisms on a -vector spaceMDLD/vector space , and let be an eigenvectorMDLD/eigenvector of and of . Show that is also an eigenvector of . What is its eigenvalue?
Exercise Create referencenumber
Let be an isomorphismMDLD/isomorphism (vs) on a -vector spaceMDLD/vector space , and let be its inverse mapping.MDLD/inverse mapping Show that is an eigenvalueMDLD/eigenvalue of if and only if is an eigenvalue of .
Exercise Create referencenumber
Give an example of a linear mappingMDLD/linear mapping
such that has no eigenvalue,MDLD/eigenvalue but a certain powerMDLD/power (endomorphism) , , has an eigenvalue.
Exercise Create referencenumber
Let be a field, and let be an endomorphismMDLD/endomorphism (linear) on a -vector spaceMDLD/vector space , satisfying
for a certain .[1] Show that every eigenvalueMDLD/eigenvalue of fulfills the property .
Exercise Create referencenumber
Let be a field, and let be an endomorphismMDLD/endomorphism (linear) on a -vector spaceMDLD/vector space . Let be an eigenvalueMDLD/eigenvalue of and a polynomial.MDLD/polynomial (1K) Show that is an eigenvalue[2] of .
Exercise Create referencenumber
Let be a square matrix, which can be written as a block matrix
with square matrices and . Show that a number is an eigenvalueMDLD/eigenvalue of if and only if is an eigenvalue of or of .
===Exercise Exercise 27.14
change===
Let be a field,MDLD/field a -vector spaceMDLD/vector space and
a linear mapping.MDLD/linear mapping Show that the following statements hold.
- Every
eigenspaceMDLD/eigenspace
is a linear subspaceMDLD/linear subspace of .
- is an eigenvalueMDLD/eigenvalue for if and only if the eigenspace is not the nullspace.MDLD/nullspace
- A vector , is an eigenvectorMDLD/eigenvector for if and only if .
Exercise Create referencenumber
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.
Exercise Create referencenumber
Let denote the real vector space, which consists of all functions from tp , which are arbitrarily often differentiable.
a) Show that the derivation is a linear mappingMDLD/linear mapping from to .
b) Determine the
eigenvaluesMDLD/eigenvalues
of the derivation and determine, for each eigenvalue, at least one
eigenvector.[3]MDLD/eigenvector
c) Determine for every real number the
eigenspaceMDLD/eigenspace
and its
dimension.MDLD/dimension (vs)
===Exercise Exercise 27.17
change===
Let be a field,MDLD/field a -vector spaceMDLD/vector space and
a linear mapping.MDLD/linear mapping Show that
Exercise Create referencenumber
Let be a field, and let be an endomorphismMDLD/endomorphism (linear) on a -vector spaceMDLD/vector space . Let and let
be the corresponding eigenspace.MDLD/eigenspace Show that can be restricted to a linear mapping
and that this mapping is the homothetyMDLD/homothety with scale factor .
===Exercise Exercise 27.19
change===
Let be a field,MDLD/field a -vector spaceMDLD/vector space and
a linear mapping.MDLD/linear mapping Let be elements in . Show that
===Exercise Exercise 27.20
change===
Let be a field,MDLD/field a finite-dimensionalMDLD/finite-dimensional (vs) -vector spaceMDLD/vector space and
a linear mapping.MDLD/linear mapping Show that there exist at most many eigenvaluesMDLD/eigenvalues for .
- Hand-in-exercises
Exercise (1 mark) Create referencenumber
Check whether the vector is an eigenvectorMDLD/eigenvector of the matrix
In this case, determine the corresponding eigenvalue.MDLD/eigenvalue
Exercise (1 mark) Create referencenumber
Check whether the vector is an eigenvectorMDLD/eigenvector of the matrix
In this case, determine the corresponding eigenvalue.MDLD/eigenvalue
Exercise (4 (1+3) marks) Create referencenumber
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) Create referencenumber
Let be a field, and let be an endomorphismMDLD/endomorphism (linear) on a -vector spaceMDLD/vector space . Show that is a homothetyMDLD/homothety if and only if every vector , , is an eigenvectorMDLD/eigenvector of .
Exercise (4 marks) Create referencenumber
Consider the matrix
Show that , as a real matrix, has no eigenvalue.MDLD/eigenvalue Determine the eigenvalues and the eigenspacesMDLD/eigenspaces of as a complexMDLD/complex (number) matrix.
Exercise (6 marks) Create referencenumber
Consider the realMDLD/real matricesMDLD/matrices
Characterize, in dependence on , when such a matrix has
- two different eigenvalues,MDLD/eigenvalues
- one eigenvalue with a two-dimensional eigenspace,MDLD/eigenspace
- one eigenvalue with a one-dimensional eigenspace,MDLD/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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