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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