Linear algebra (Osnabrück 2024-2025)/Part II/Exercise sheet 42

Exercises

Exercise

A complex number ${}z\in \mathbb {C}$ defines an endomorphism ${}\mu _{z}\colon x\rightarrow zx$. Sketch, in the plane ${}\mathbb {C}$, the complex numbers with the property that ${}\mu _{z}$ is an isometry, self-ajoint, a self-adjoint isometry, normal.

Exercise

A complex number ${}z\in \mathbb {C}$ defines a homothety ${}\mu _{z}\colon v\rightarrow zv$ on ${}\mathbb {C} ^{n}$. For what numbers ${}z$ is this an isometry, a self-adjoint endomorphism, a normal endomorphism?

Exercise

When is a shear mapping ${}{\begin{pmatrix}1&a\\0&1\end{pmatrix}}$ on ${}\mathbb {R} ^{2}$ a normal endomorphism?

Exercise

Let ${}V$ be a finite-dimensional ${}{\mathbb {K} }$-vector space, endowed with an inner product ${}\left\langle -,-\right\rangle$, and let

$\varphi \colon V\longrightarrow V$

denote an endomorphism. Show that ${}\varphi$ is normal if and only if the adjoint endomorphism ${}{\hat {\varphi }}$ is normal.

Exercise

Let ${}V$ be a finite-dimensional ${}{\mathbb {K} }$-vector space, endowed with an inner product, and let

${}V=V_{1}\oplus V_{2}\,$

denote a direct sum decomposition into linear subspaces ${}V_{1}$ and ${}V_{2}$, which are orthogonal to each other. Let

$\varphi _{1}\colon V_{1}\longrightarrow V_{1}$

and

$\varphi _{2}\colon V_{2}\longrightarrow V_{2}$

be normal endomorphisms, and

${}\varphi =\varphi _{1}\oplus \varphi _{2}\,$

their direct sum. Show that also ${}\varphi$ is normal.

Exercise

Let ${}V$ be a finite-dimensional ${}{\mathbb {K} }$-vector space, endowed with an inner product ${}\left\langle -,-\right\rangle$. Let

$\varphi \colon V\longrightarrow V$

denote a normal endomorphism. Show

${}\operatorname {kern} \varphi =\operatorname {kern} {\hat {\varphi }}\,.$

To solve the following exercise, use Theorem 42.9 and Exercise 26.22 .

Exercise

Let ${}V$ be a ${}{\mathbb {K} }$-vector space, endowed with an inner product. Let

$\varphi \colon V\longrightarrow V$

be a normal endomorphism, and let ${}U\subseteq V$ denote a ${}\varphi$-invariant linear subspace. Show that ${}U$ is also invariant under the adjoint endomorphism ${}{\hat {\varphi }}$.

Exercise

Let ${}V$ be a ${}{\mathbb {K} }$-vector space, endowed with an inner product. Let

$\varphi \colon V\longrightarrow V$

be a normal endomorphism, and let ${}U\subseteq V$ denote a ${}\varphi$-invariant linear subspace. Show that also the restriction

$\varphi {|}_{U}\colon U\longrightarrow U$

is normal.

Exercise

Let ${}V$ be a ${}{\mathbb {K} }$-vector space, endowed with an inner product ${}\left\langle -,-\right\rangle$. Show that the set of all normal endomorphisms of ${}V$ is not a linear subspace in ${}\operatorname {End} _{}{\left(V\right)}$.

Exercise

We consider the linear mapping ${}\varphi \colon \mathbb {R} ^{2}\rightarrow \mathbb {R} ^{2}$ given by the matrix ${}{\begin{pmatrix}7&0\\0&7\end{pmatrix}}$ with respect to the basis ${}{\begin{pmatrix}6\\-5\end{pmatrix}},{\begin{pmatrix}-5\\2\end{pmatrix}}$. Is this a normal endomorphism?

Exercise

We consider the linear mapping ${}\varphi \colon \mathbb {R} ^{2}\rightarrow \mathbb {R} ^{2}$ given by the matrix ${}{\begin{pmatrix}5&0\\0&6\end{pmatrix}}$ with respect to the basis ${}{\begin{pmatrix}7\\-2\end{pmatrix}},{\begin{pmatrix}4\\14\end{pmatrix}}$. Is this a normal endomorphism?

Exercise

Determine whether there exists, for the linear mapping ${}\mathbb {C} ^{2}\rightarrow \mathbb {C} ^{2}$ given by the matrix

${\begin{pmatrix}2-3{\mathrm {i} }&4+5{\mathrm {i} }\\11-3{\mathrm {i} }&6+9{\mathrm {i} }\end{pmatrix}},$

an orthonormal basis of ${}\mathbb {C} ^{2}$ consisting of eigenvectors.

Exercise

Determine whether there exists, for the linear mapping ${}\mathbb {C} ^{3}\rightarrow \mathbb {C} ^{3}$ given by the matrix

${\begin{pmatrix}2-5{\mathrm {i} }&3-4{\mathrm {i} }&1+5{\mathrm {i} }\\0&1-{\mathrm {i} }&2+9{\mathrm {i} }\\6+3{\mathrm {i} }&{\mathrm {i} }-7&-4{\mathrm {i} }\end{pmatrix}},$

an orthonormal basis of ${}\mathbb {C} ^{3}$ consisting of eigenvectors.

Exercise

Let ${}V$ be a finite-dimensional ${}{\mathbb {K} }$-vector space, endowed with an inner product ${}\left\langle -,-\right\rangle$, and let ${}v_{1},\ldots ,v_{n}$ denote a basis of ${}V$. Let

$\varphi \colon V\longrightarrow V$

be an endomorphism, and let ${}\Psi _{\varphi }$ denote the corresponding sesquilinear form in the sense of Lemma 41.12 . What is the relation between the describing matrix of ${}\varphi$ and the Gram matrix of ${}\Psi _{\varphi }$? What is the relation with the Gram matrix of the form ${}\Theta _{\varphi }$, defined by

${}\Theta _{\varphi }(v,w)=\left\langle v,\varphi (w)\right\rangle \,?$

Exercise

Let ${}X^{2}+(3-2{\mathrm {i} })X-6{\mathrm {i} }$ be the characteristic polynomial of a normal endomorphism ${}\varphi \colon \mathbb {C} ^{2}\rightarrow \mathbb {C} ^{2}$. Determine the characteristic polynomial of the adjoint endomorphism ${}{\hat {\varphi }}$.

Exercise

Let ${}V$ be a finite-dimensional ${}{\mathbb {K} }$-vector space, with a fixed inner product ${}\left\langle -,-\right\rangle$. We call a sesquilinear form ${}\Psi$ on ${}V$ orthogonalizable if there exsts an orthonormal basis ${}u_{1},\ldots ,u_{n}$ (with respect to the inner product) of ${}V$ fulfilling

${}\Psi (u_{i},u_{j})=0\,$

for all ${}i\neq j$. Show that via the correspondence

$\operatorname {End} _{}{\left(V\right)}\longrightarrow \operatorname {Sesq} _{}{\left(V\right)},\varphi \longmapsto \Psi _{\varphi },$

the normal endomorphisms correspond to the orthogonalizable sesquilinear forms.

Exercise

Let ${}V$ be a finite-dimensional ${}{\mathbb {K} }$-vector space, and let ${}\left\langle -,-\right\rangle$ denote an hermitian sesquilinear form on ${}V$. Show that there exists an orthogonal basis on ${}V$.

Exercise

Prove the inertia law of Sylvester for a complex-hermitian form.

Exercise

Suppose that ${}\mathbb {R} ^{4}$ is endowed (beside the standard inner product) with the standard Minkowski form. Give a basis of ${}\mathbb {R} ^{4}$ that is an orthonormal basis with respect to the inner product, and an orthogonal basis with respect to the Minkowski form.

Exercise

Suppose that ${}\mathbb {R} ^{2}$ is endowed (beside the standard inner product) with the standard Minkowski form. Determine all bases of ${}\mathbb {R} ^{2}$ that are an orthonormal basis with respect to the inner product and an orthogonal basis with respect to the Minkowski form.

Exercise

Determine the type of the symmetric bilinear form given by the Gram matrix

${\begin{pmatrix}-4&7&3\\7&-9&-2\\3&-2&1\end{pmatrix}}.$

Exercise

Determine the type of the matrix

${\begin{pmatrix}-6&3-2{\mathrm {i} }&-1+3{\mathrm {i} }\\3+2{\mathrm {i} }&1&5\\-1-3{\mathrm {i} }&5&-1\end{pmatrix}}.$

Exercise

Let ${}\Psi$ be an hermitian form with the Gram matrix ${}G$ (with respect to a given basis). Show that the determinant of ${}G$ is real.

Exercise

Let ${}\Psi$ be an hermitian form with the Gram matrix ${}G$ (with respect to a given basis). Show that the characteristic polynomial of ${}G$ has real coefficients.

Hand-in-exercises

Exercise (2 marks)

We consider the linear mapping ${}\varphi \colon \mathbb {R} ^{2}\rightarrow \mathbb {R} ^{2}$ given by the matrix ${}{\begin{pmatrix}3&0\\0&5\end{pmatrix}}$ with respect to the basis ${}{\begin{pmatrix}7\\-2\end{pmatrix}},{\begin{pmatrix}-4\\3\end{pmatrix}}$. Is this a normal endomorphism?

Exercise (2 marks)

Let ${}V$ be a finite-dimensional ${}{\mathbb {K} }$-vector space, endowed with an inner product, and let

$\varphi \colon V\longrightarrow V$

denote a normal endomorphism. Show that also ${}\varphi -\lambda \cdot \operatorname {Id} _{V}$ is normal.

Exercise (3 marks)

Determine whether there exists, for the linear mapping ${}\mathbb {C} ^{3}\rightarrow \mathbb {C} ^{3}$ given by the matrix

${\begin{pmatrix}4+{\mathrm {i} }&6-7{\mathrm {i} }&6+3{\mathrm {i} }\\4-{\mathrm {i} }&2-{\mathrm {i} }&2-{\mathrm {i} }\\5+3{\mathrm {i} }&2{\mathrm {i} }-11&4+3{\mathrm {i} }\end{pmatrix}},$

an orthonormal basis of ${}\mathbb {C} ^{3}$ consisting of eigenvectors.

Exercise (4 marks)

Let

$\varphi \colon V\longrightarrow V$

be a normal endomorphism on the finite-dimensional ${}\mathbb {C}$-vector space ${}V$. Show that ${}\varphi$ is self-adjoint if and only if all eigenvalues of ${}\varphi$ are real.

Exercise (3 marks)

Determine the type of the matrix

${\begin{pmatrix}-5&3-{\mathrm {i} }&-4+{\mathrm {i} }\\3+{\mathrm {i} }&2&7{\mathrm {i} }\\-4-{\mathrm {i} }&-7{\mathrm {i} }&4\end{pmatrix}}.$

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