Linear algebra (Osnabrück 2024-2025)/Part I/Exercise sheet 24



Exercise for the break

Confirm the Theorem of Cayley-Hamilton for the matrix

by an explicit computation.




Exercises

Confirm the Theorem of Cayley-Hamilton for the matrix

by an explicit computation.


Confirm the Theorem of Cayley-Hamilton for an upper triangular matrix of the form


Let be a diagonalizable matrix with the characteristic polynomial . Show directly that

holds.


Let denote a field, and let denote a -vector space of finite dimension. Let

be a linear mapping. Let

be the -fold direct sum of with itself. How is the minimal polynomial (the characteristic polynomial) of related to the minimal polynomial (the characteristic polynomial) of ?


Express the matrix

(whose entries are in ) in the form

with matrices .


Let be an -matrix over a field , and suppose that its minimal polynomial has the form

with different . Show that is diagonalizable.


Let be a field, let , and let be the set of all -th roots of unity in . Show that is a subgroup of the unit group .


Show that every complex root of unity lies on the unit circle.


An -th root of unity is called primitive, if its order

is .

Let be an -th primitive root of unity in a field . Show the formula


Let be the permutation matrix for a transposition. Show that is diagonalizable over .


Let the cycle be given, and let denote the corresponding -permutation matrix over a field .

a) Let be a polynomial of degree . Establish a formula for .


b) Determine the minimal polynomial of .


c) Give an example for an endomorphism on a real vector space with different vectors such that , and holds, and such that the minimal polynomial of is not .


Suppose that, for some permutation , its cycle decomposition is known. Determine the minimal polynomial and the characteristic polynomial of the permutation matrix .


Let be a permutation, and let denote the corresponding permutation matrix over a field . For , let


a) Show that is -invariant if and only if .


b) Show that there might exist -invariant subspaces that are not of the form .


Let be a finite field. Show that every unit in is a root of unity.


Determine the order of the matrix

over the field with elements.


Let be a finite field, and let denote an invertible -matrix over . Show that has finite order.


Give a matrix of order .




Hand-in-exercises

Exercise (4 marks)

Confirm the Theorem of Cayley-Hamilton for the matrix

by an explicit computation.


Exercise (4 marks)

Let be an -matrix over a field , and let

be a polynomial with

and with . Show that is invertible, and that its inverse matrix is given by


Exercise (5 marks)

Let and be finite-dimensional -vector spaces, and let

and

be endomorphisms, with the minimal polynomials and . Show that the minimal polynomial of

equals the normed generator of the ideal .


Exercise (3 marks)

Determine the order of the matrix

over the field with elements.


Exercise (4 marks)

Show that a permutation matrix over is diagonalizable.



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