Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 28/refcontrol
- Exercises
Exercise Create referencenumber
Compute the characteristic polynomialMDLD/characteristic polynomial of the matrixMDLD/matrix
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Compute the characteristic polynomial,MDLD/characteristic polynomial the eigenvaluesMDLD/eigenvalues and the eigenspacesMDLD/eigenspaces of the matrix
over .
===Exercise Exercise 28.3
change===
Show that the characteristic polynomial of a linear mappingMDLD/linear mapping on a finite-dimensionalMDLD/finite-dimensional (vs) -vector spaceMDLD/vector space is well-defined, that is, independent of the chosen basis.MDLD/basis (vs)
===Exercise Exercise 28.4
change===
Let be a fieldMDLD/field and let denote an -matrixMDLD/matrix over . Show that for every , the relation
holds.[1]
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Let be a field and let be an -matrixMDLD/matrix over . Where can you find the determinantMDLD/determinant of within the characteristic polynomialMDLD/characteristic polynomial ?
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Show that the characteristic polynomialMDLD/characteristic polynomial of the so-called Companion matrix
equals
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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 .
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Let
- Determine the characteristic polynomialMDLD/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.
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Let be a zero of the polynomial
Show that
is an eigenvectorMDLD/eigenvector of the matrix
for the eigenvalueMDLD/eigenvalue .
To solve the following exercise, the two exercises above and also
Exercise 24.31
are helpful.
===Exercise Exercise 28.10
change===
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.
Exercise Create referencenumber
Determine the eigenvalues and the eigenspaces of the linear mapping
given by the matrix
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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.
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Let
Compute:
- the eigenvalues of ;
- the corresponding eigenspaces;
- the geometric and algebraic multiplicities of each eigenvalue;
- a matrix such that is a diagonal matrix.
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Determine the eigenspaceMDLD/eigenspace and the geometric multiplicityMDLD/geometric multiplicity for of the matrix
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Show that the matrixMDLD/matrix
is diagonalizableMDLD/diagonalizable over .
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Let be a matrixMDLD/matrix with (pairwise) different eigenvalues.MDLD/eigenvalues Show that the determinantMDLD/determinant of is the product of the eigenvalues.
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Let be a field, and numbers with . Give an example of an -matrixMDLD/matrix , such that is an eigenvalueMDLD/eigenvalue for with algebraic multiplicityMDLD/algebraic multiplicity and geometric multiplicityMDLD/geometric multiplicity .
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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.
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Determine, whether the real matrix
is trigonalizableMDLD/trigonalizable or not.
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Suppose that a linear mappingMDLD/linear mapping
is given by the matrixMDLD/matrix
with respect to the standard basis.MDLD/standard basis Find a basis,MDLD/basis (vs) 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.MDLD/linear mapping A linear subspaceMDLD/linear subspace is called -invariant, if
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Let a linear mappingMDLD/linear mapping on a -vector spaceMDLD/vector space over a field . Show the following properties.
- The zero spaceMDLD/zero space is -invariant.MDLD/invariant (subspace)
- is -invariant.
- EigenspacesMDLD/Eigenspaces are -invariant.
- Let be -invariant linear subspaces. Then also and are -invariant.
- Let be a -invariant linear subspace. Then also the image spaceMDLD/image space and the preimage spaceMDLD/preimage space are -invariant.
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Let a linear mappingMDLD/linear mapping on a -vector spaceMDLD/vector space over a field , and let . Show that the smallest -invariant linear subspaceMDLD/invariant linear subspace of , which contains , equals
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Let a linear mappingMDLD/linear mapping on a -vector spaceMDLD/vector space over a field . Show that the subset of , defined by
is an -invariant linear subspace.MDLD/invariant linear subspace
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Let be a linear mappingMDLD/linear mapping on a -vector spaceMDLD/vector space . Let be a basisMDLD/basis (vs) of , such that is described, with respect to this basis, by an upper triangular matrix.MDLD/upper triangular matrix Show that the linear subspacesMDLD/linear subspaces
are -invariantMDLD/invariant (endomorphism) for every .
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Determine, whether the real matrix
is trigonalizableMDLD/trigonalizable or not.
- Hand-in-exercises
Exercise (2 marks) Create referencenumber
Compute the characteristic polynomialMDLD/characteristic polynomial of the matrixMDLD/matrix
Exercise (3 marks) Create referencenumber
Compute the characteristic polynomial,MDLD/characteristic polynomial the eigenvaluesMDLD/eigenvalues and the eigenspacesMDLD/eigenspaces of the matrix
over .
Exercise (4 marks) Create referencenumber
Let
be a linear mapping.MDLD/linear mapping Show that has at least one eigenvector.MDLD/eigenvector
Exercise (4 marks) Create referencenumber
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) Create referencenumber
Determine for every the algebraicMDLD/algebraic (multiplicity) and geometricMDLD/geometric (multiplicity) multiplicities for the matrixMDLD/matrix
Exercise (4 marks) Create referencenumber
Decide whether the matrixMDLD/matrix
is trigonalizableMDLD/trigonalizable over .
Exercise (3 marks) Create referencenumber
Determine whether the real matrix
is trigonalizableMDLD/trigonalizable or not.
- Footnotes
- ↑ The main difficulty might be here to recognize that there is indeed something to show.
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