Math Problems Vault

Problem 14.1.1

(i)(ii)

Let $A$ be a nonnormal $n \times n$ matrix over $\mathbb {C}$ .

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(i)

Show that $A+A^{*}$ is a normal matrix.

(ii)

Show that $AA^{*}$ is a normal matrix.

Problem 14.1.2

Let $A$ be a nonnormal matrix. Show that $[A,A^{*}]$ is a normal matrix. Show that $[A,A^{*}]_{+}$ is a normal matrix.

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Problem 14.1.3

Let $\alpha , \beta \in \mathbb {C}$ . What are the conditions on $\alpha$ , $\beta$ such that the matrix

M(\alpha ,\beta ) = \begin{pmatrix} \alpha & \beta \\ 0 & 1 \end{pmatrix} \Rightarrow M^{*}(\alpha ,\beta ) = \begin{pmatrix} \bar\alpha & 0 \\ \bar\beta & 1 \end{pmatrix}

is normal?

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Problem 14.1.4

Let $\theta \in [0,2\pi )$ . What are the conditions on $\theta$ such that the matrix $A(\theta )$

A(\theta ) = \begin{pmatrix} 0 & \sin (\theta ) \\ \cos (\theta ) & 0 \end{pmatrix} \Rightarrow A^{*}(\theta ) = \begin{pmatrix} 0 & \cos (\theta ) \\ \sin (\theta ) & 0 \end{pmatrix}

is normal?

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Problem 14.1.5

Let $\theta \in \mathbb {R}$ . What are the conditions on $\theta$ such that the matrix $A(\theta )$

A(\theta ) = \begin{pmatrix} 0 & \sinh (\theta ) \\ \cosh (\theta ) & 0 \end{pmatrix} \Rightarrow A^{*}(\theta ) = \begin{pmatrix} 0 & \cosh (\theta ) \\ \sinh (\theta ) & 0 \end{pmatrix}

is normal?

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Problem 14.1.6

Let $z \in \mathbb {C}$ and $z \neq 0$ . Show that the $2 \times 2$ matrix

A = \begin{pmatrix} 1 & z \\ 0 & -1 \end{pmatrix}

is nonnormal. Give the eigenvalues.

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Problem 14.1.7

(i)(ii)

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(i)

Can we conclude that an invertible matrix $A$ is normal?

(ii)

Let $B$ be an $n \times n$ matrix over $\mathbb {C}$ with $B^{2} = I_{n}$ . Can we conclude that $B$ is normal?

Problem 14.1.8

Show that not all nonnormal matrices are non-diagonalizable.

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Problem 14.1.9

Find all $2 \times 2$ matrices over $\mathbb {C}$ which are nonnormal but diagonalizable.

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Problem 14.1.10

Let $A$ be a normal $n \times n$ matrix and let $S$ be an invertible nonnormal matrix. Is $SAS^{-1}$ a normal matrix? Study the case

A = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}, \quad S = \begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix}.

Of course the eigenvalues of $A$ and $SAS^{-1}$ are the same.

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Problem 14.1.11

Consider the nonnormal $2 \times 2$ matrix

A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}.

Find $\sinh (A)$ and $\tanh^{-1}(A)$ .

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Problem 14.1.12

(i)(ii)

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(i)

Consider the nonnormal $2 \times 2$ matrix

A = \begin{pmatrix} 0 & i \\ 0 & 0 \end{pmatrix} \Rightarrow A^{*} = \begin{pmatrix} 0 & 0 \\ -i & 0 \end{pmatrix}.

Find a unitary matrix $U$ such that $UAU^{*} = A^{*}$ .

(ii)

Consider the nonnormal $2 \times 2$ matrix

K = \begin{pmatrix} -i & i \\ 0 & i \end{pmatrix}.

Find a unitary matrix $V$ such $VKV^{*} = K^{*}$ . Find the eigenvalues and normalized eigenvectors of $K$ and $K^{*}$ . Discuss.

Problem 14.1.13

Let $A$ , $B$ be normal matrices. Can we conclude that $AB$ is a normal matrix?

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Problem 14.1.14

Consider the nonnormal $3 \times 3$ matrix

B = \begin{pmatrix} 0 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \end{pmatrix}.

Can we find a $3 \times 3$ matrix $Y$ such that $Y^{2} = B$ , i.e. $Y$ would be the square root of $B$ .

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Problem 14.1.15

Let $A$ be a normal $n \times n$ matrix. Is the matrix $A-iI_{n}$ normal?

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Problem 14.1.16

A measure of nonnormality is given by

m(A) := \left\| A^{*}A-AA^{*}\right\|

where $\left\| \cdot \right\|$ denotes some matrix norm. Let $S_{3}$ and $S_{1}$ be real-valued and symmetric $n \times n$ matrices (for example, spin matrices). Calculate $m(S_{3}+\mathrm{exp}\left(i\phi \right)S_{1})$ .

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Problem 14.1.17

Consider the nonnormal matrix

A = \begin{pmatrix} i\pi & 1 \\ 0 & -i\pi \end{pmatrix}.

Is $e^{A+A^{*}} = e^{A}e^{A^{*}}$ ?

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Problem 14.1.18

(i)(ii)

Prove or disprove the following statements.

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(i)

If the $n \times n$ matrix $A$ is nonnormal, then there exists no matrix $B$ such that $B^{2} = A$ .

(ii)

Let $A$ , $B$ be nonzero $n \times n$ matrices with $B^{2} = A$ . Then the matrix $A$ is normal.

Problem 14.1.19

We know that all real symmetric matrices are diagonalizable. Are all complex symmetric matrices are diagonalizable?

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Problem 14.1.20

Let $A$ be a nonnormal $n \times n$ matrix and let $U$ be a unitary $n \times n$ matrix. Is $\widetilde A = UAU^{-1}$ nonnormal for all $U$ ?

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Problem 14.1.21

Let $A$ be a nonnormal matrix. Show that $A^{*}A$ and $AA^{*}$ are linearly independent.

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Problem 14.1.22

(i)(ii)

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(i)

Show that if $A$ and $B$ are nonnormal, then $A \otimes B$ is nonnormal.

(ii)

Show that if $A$ and $B$ are nonnormal, then $A \oplus B$ is nonnormal, where $\oplus$ denotes the direct sum.