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

Exercise for the break

Exercise

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

${\begin{pmatrix}2&0&\cdots &\cdots &\cdots &0\\0&1&\ddots &\ddots &\vdots &\vdots \\\vdots &\ddots &3&\ddots &\ddots &\vdots \\\vdots &\ddots &\ddots &1&\ddots &\vdots \\\vdots &\vdots &\ddots &\ddots &3&0\\0&\cdots &\cdots &\cdots &0&1\end{pmatrix}}.$

Exercises

Exercise

Determine the eigenspace and the geometric multiplicity for ${}-2$ of the matrix

${\begin{pmatrix}2&1&3\\5&0&7\\9&3&8\end{pmatrix}}.$

Exercise

Let ${}K$ be a field, ${}V$ a finite-dimensional ${}K$-vector space and

$\varphi \colon V\longrightarrow V$

a linear mapping. Show that there exist at most ${}\dim _{K}{\left(V\right)}$ many eigenvalues for ${}\varphi$.

Exercise

For a given ${}a\in K$ and ${}m$, ${}1\leq m\leq n$, describe an ${}n\times n$-matrix over ${}K$ such that ${}a$ is its only eigenvalue with geometric multiplicity ${}m$.

Exercise

Let ${}M\in \operatorname {Mat} _{n}(K)$ be a matrix with ${}n$ (pairwise) different eigenvalues. Show that the determinant of ${}M$ is the product of the eigenvalues.

Exercise

Let

${}M\in \operatorname {Mat} _{n}(K)\,$

be a matrix with ${}n$ different eigenvalues. Show that the trace of ${}M$ is the sum of these eigenvalues.

Exercise

Show that the matrix

${\begin{pmatrix}0&1\\1&0\end{pmatrix}}$

is diagonalizable over ${}\mathbb {Q}$.

Exercise

Let ${}K$ denote a field, and let ${}V$ denote a ${}K$-vector space of finite dimension. Let

$\varphi \colon V\longrightarrow V$

be a linear mapping, and let ${}\lambda \in K$. Show that the exponent of ${}X-\lambda$ in the minimal polynomial of ${}\varphi$ might be smaller, but also larger, than the geometric multiplicity of ${}\lambda$.

Exercise

Show that the matrix

${\begin{pmatrix}2&3\\0&5\end{pmatrix}}$

is diagonalizable over ${}\mathbb {R}$, and determine a basis consisting of eigenvectors. Perform the base change, yielding the describing diagonal matrix.

Exercise

Show that the matrix

${\begin{pmatrix}6&1&0\\0&2&4\\0&0&7\end{pmatrix}}$

is diagonalizable over ${}\mathbb {R}$, and determine a basis consisting of eigenvectors. Perform the base change, yielding the describing diagonal matrix.

Exercise

Let ${}M$ be an invertible matrix over ${}K$. Show that ${}M$ is diagonalizable if and only if the inverse matrix ${}M^{-1}$ is diagonalizable.

Exercise

Determine which elementary matrices are diagonalizable.

Exercise

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

Exercise

Show that a projection is diagonalizable.

Exercise

Let ${}\varphi \colon V\rightarrow V$ be a diagonalizable endomorphism on the finite-dimensional ${}K$-vector space ${}V$, and let ${}P\in K[X]$ be a polynomial. Show that ${}P(\varphi )$ is also diagonalizable.

Exercise

Let ${}M$ be a diagonalizable matrix. Show that the minimal polynomial of ${}M$ has the form

$(X-\lambda _{1})\cdots (X-\lambda _{k})$

with different ${}\lambda _{i}$.

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

Hand-in-exercises

Exercise (4 marks)

Let ${}V_{1},\ldots ,V_{n}$ be finite-dimensional vector spaces over the field ${}K$, and let

$\varphi _{i}\colon V_{i}\longrightarrow V_{i}$

be linear mappings. Let

$\varphi =\varphi _{1}\times \cdots \times \varphi _{n}\colon V_{1}\times \cdots \times V_{n}\longrightarrow V_{1}\times \cdots \times V_{n}$

be the product mapping. Show that ${}\varphi$ is diagonalizable if and only if all ${}\varphi _{i}$ are diagonalizable.

Exercise (3 marks)

Let ${}K$ denote a field, and let ${}V$ denote a ${}K$-vector space of finite dimension. Let

$\varphi \colon V\longrightarrow V$

be a linear mapping. Let

${\varphi }^{*}\colon {V}^{*}\longrightarrow {V}^{*}$

denote the dual mapping. Show that ${}\varphi$ is diagonalizable if and only if ${}{\varphi }^{*}$ is diagonalizable.

Exercise (4 marks)

Show that the matrix

${\begin{pmatrix}0&1\\-1&0\end{pmatrix}}$

is diagonalizable over ${}\mathbb {C}$, but not over ${}\mathbb {R}$. Perform the diagonalization over ${}\mathbb {C}$.

Exercise (4 marks)

Show that the matrix

${\begin{pmatrix}0&1\\1&0\end{pmatrix}}$

over the field with two elements ${}{\mathbb {F} }_{2}$ is not diagonalizable.

<< | Linear algebra (Osnabrück 2024-2025)/Part I | >> PDF-version of this exercise sheet Lecture for this exercise sheet (PDF)