Linear algebra (Osnabrück 2024-2025)/Part I/Exercise sheet 21



Exercise for the break

Check whether the vector is an eigenvector for the matrix

In this case, determine the corresponding eigenvalue.




Exercises

Let a linear mapping

be given. What are the eigenvalues and the eigenvectors of ?


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?


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 .


Give an example of a linear mapping

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


Show that every matrix has at least one eigenvalue.


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 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 .


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 of .


Let denote vector spaces over a field , and let

denote linear mappings. Let be an eigenvalue of for a determined . Show that is also an eigenvalue of the product mapping


Show that is an eigenvalue for the linear mapping

given by a matrix of the form if and only if is a zero of the polynomial


The concept of an eigenvector is also defined for infinite-dimensional vector spaces, and it is also important in this context, as the following exercise shows.

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.[1]


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


Let

be an endomorphism on a finite-dimensional -vector space , and let be an eigenvector for with eigenvalue . Let

be the dual mapping of . We consider bases of of the form with the dual basis . Give examples of the following behavior.

a) is an eigenvector of with the eigenvalue independent of .


b) is an eigenvector of with the eigenvalue with respect to some basis , but not with respect to another basis .


c) is for no basis an eigenvector of .


Let be a finite-dimensional -vector space, and , , a fixed vector. Show that

with the natural addition and multiplication of endomorphisms, is a ring and a linear subspace of . Determine the dimension of this space.


Let be a field, and let denote a vector different from . Establish an inhomogeneous system of linear equations such that its solution set is the set of all -matrices, for that is an eigenvector with eigenvalue . What is special about this system, and what is the dimension of its solution set?


Let be a real -matrix. Let be a real number, and suppose that it is an eigenvalue of , considered as a complex matrix. Show that is already over an eigenvalue of .

Generalize the previous statement for any field extension .



Hand-in-exercises

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

  1. two different eigenvalues,
  2. one eigenvalue with a two-dimensional eigenspace,
  3. one eigenvalue with a one-dimensional eigenspace,
  4. no eigenvalue.


Exercise (2 marks)

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

for a certain .[2] Show that every eigenvalue of fulfills the property .


Exercise (4 marks)

Let be a field, and let denote an -dimensional -vector space. Let

be a linear mapping. Let be an eigenvalue of , and a corresponding eigenvector. Show that, for a given basis of , there exists a basis such that and such that

for all holds.

Show also that this is not possible for .




Footnotes
  1. In this context, one also says eigenfunction instead of eigenvector.
  2. The value is allowed, but does not say much.


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