Linear algebra (Osnabrück 2024-2025)/Part I/Exercise sheet 22



Exercise for the break

Determine the eigenvalues, the eigenspaces, and the geometric multiplicities of the matrix




Exercises

Determine the eigenspace and the geometric multiplicity for of the matrix


Let be a field, a finite-dimensional -vector space and

a linear mapping. Show that there exist at most many eigenvalues for .


For a given and , , describe an -matrix over such that is its only eigenvalue with geometric multiplicity .


Let be a matrix with (pairwise) different eigenvalues. Show that the determinant of is the product of the eigenvalues.


Let

be a matrix with different eigenvalues. Show that the trace of is the sum of these eigenvalues.


Show that the matrix

is diagonalizable over .


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

be a linear mapping, and let . Show that the exponent of in the minimal polynomial of might be smaller, but also larger, than the geometric multiplicity of .


Show that the matrix

is diagonalizable over , and determine a basis consisting of eigenvectors. Perform the base change, yielding the describing diagonal matrix.


Show that the matrix

is diagonalizable over , and determine a basis consisting of eigenvectors. Perform the base change, yielding the describing diagonal matrix.


Let be an invertible matrix over . Show that is diagonalizable if and only if the inverse matrix is diagonalizable.


Determine which elementary matrices are diagonalizable.


Let be an upper triangular matrix, where all diagonal entries equal . Show that is diagonalizable if and only if it is already a diagonal matrix.


Show that a projection is diagonalizable.


Let be a diagonalizable endomorphism on the finite-dimensional -vector space , and let be a polynomial. Show that is also diagonalizable.


Let be a diagonalizable matrix. Show that the minimal polynomial of has the form

with different .

The reverse statement of the preceding exercise is also true, see Exercise 24.7 .



Hand-in-exercises

Exercise (4 marks)

Let be finite-dimensional vector spaces over the field , and let

be linear mappings. Let

be the product mapping. Show that is diagonalizable if and only if all are diagonalizable.


Exercise (3 marks)

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

be a linear mapping. Let

denote the dual mapping. Show that is diagonalizable if and only if is diagonalizable.


Exercise (4 marks)

Show that the matrix

is diagonalizable over , but not over . Perform the diagonalization over .


Exercise (4 marks)

Show that the matrix

over the field with two elements is not diagonalizable.



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