Math Problems Vault

Problem 14.2.1

Are all nonzero nilpotent matrices nonnormal?

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Problem 14.2.2

(i)(ii)(iii)

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(i)

Let $a_{11}, a_{22}, \epsilon \in \mathbb {R}$ . What is the condition on $a_{11}$ , $a_{22}$ , $\epsilon$ such that the $2 \times 2$ matrix

A = \begin{pmatrix} a_{11} & e^{\epsilon } \\ e^{-\epsilon } & a_{22} \end{pmatrix}

is normal?

(ii)

Let $a > 0$ , $b \geq 0$ and $\phi \in [0,\pi ]$ . What are the conditions $a$ , $b$ , $\phi$ such that

B(a,b,\phi ) = \begin{pmatrix} 0 & a \\ e^{i\phi }b & 0 \end{pmatrix}

is a normal matrix?

(iii)

Let $z \in \mathbb {C}$ . Is the $2 \times 2$ matrix

\begin{pmatrix} 1 & 0 \\ 1-e^{z} & e^{-z} \end{pmatrix}

nonnormal?

Problem 14.2.3

(i)(ii)

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(i)

Let $x_{1}, x_{2}, x_{3} \in \mathbb {R}$ . What is the condition such that the $3 \times 3$ matrix

A(x_{1},x_{2},x_{3}) = \begin{pmatrix} 0 & x_{1} & 0 \\ 0 & 0 & x_{2} \\ x_{3} & 0 & 0 \end{pmatrix}

is normal? Show that $\lambda^{3} = x_{1}x_{2}x_{3}$ for $\operatorname {det}\left(A-\lambda I_{3}\right) = 0$ . Find the eigenvalues and normalized eigenvectors of $A$ .

(ii)

Let $x \in \mathbb {R}$ . For which values of $x$ is the $3 \times 3$ matrix

A(x) = \begin{pmatrix} -1+ix & ix & ix \\ ix & ix & ix \\ ix & ix & 1+ix \end{pmatrix}

nonnormal.

Problem 14.2.4

Show that if $A$ is nonnormal, then $A \bullet A^{*}$ is normal, where $\bullet$ denotes the Hadamard product. Notice that $A \bullet B = B \bullet A$ and $(A \bullet B)^{*} = A^{*} \bullet B^{*}$ .

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Problem 14.2.5

Find all nonnormal $2 \times 2$ matrices $A$ such that $AA^{*}+A^{*}A = I_{2}$ . An example is

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

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Problem 14.2.6

(i)(ii)

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(i)

Can one find a nonnormal matrix $A$ such that $e^{A}e^{A^{*}} = e^{A^{*}}e^{A}$ ?

(ii)

Can one find a nonnormal matrix $A$ such that $e^{A}e^{A^{*}} = e^{A+A^{*}}$ ?

Problem 14.2.7

Let $M$ be an invertible $n \times n$ matrix. Then we can form

M_{+} = \frac{1}{2}(M+M^{-1}), \quad M_{-} = \frac{1}{2}(M-M^{-1}).

Consider the nonnormal matrix

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

Find $M_{+}$ and $M_{-}$ . Are the matrices $M_{+}$ and $M_{-}$ normal?

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Problem 14.2.8

Let $s_{j} := 2\sin (2\pi j/5)$ with $j = 1, \ldots , 5$ . Consider the $5 \times 5$ matrix

M = \begin{pmatrix} s_{1} & 1 & 0 & 0 & -1 \\ -1 & s_{2} & 1 & 0 & 0 \\ 0 & -1 & s_{3} & 1 & 0 \\ 0 & 0 & -1 & s_{4} & 1 \\ 1 & 0 & 0 & -1 & s_{5} \end{pmatrix}.

Show that the matrix is nonnormal. Find the eigenvalues and eigenvectors. Is the matrix diagonalizable?

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Problem 14.2.9

Let $A$ , $B$ be nonzero $n \times n$ hermitian matrices. We form the matrix $K$ as $K := A+iB$ . This matrix is nonnormal if $[A,B] \neq 0_{n}$ and normal if $AB = BA$ . In the following we assume that $K$ is nonnormal. Assume that we can find a unitary $n \times n$ matrix $U$ such that $UKU^{*} = K^{*}$ . What can be said about the eigenvalues of $K$ ? In physics such a unitary matrix is called a quasi-symmetry operator.

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Problem 14.2.10

Let $Q$ be a nonnormal invertible $n \times n$ matrix. Is $Q \otimes Q^{-1}$ nonnormal?

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Problem 14.2.11

Let $M$ be an $n \times n$ nonnormal matrix. Are the $2n \times 2n$ matrices

A = \begin{pmatrix} 0_{n} & M^{*} \\ M & 0_{n} \end{pmatrix}, \quad B = \begin{pmatrix} 0_{n} & -M^{*} \\ M & 0_{n} \end{pmatrix}

nonnormal?

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Problem 14.2.12

Let $A$ be a nonnormal invertible matrix. How do we construct a matrix $B$ such that $A = \mathrm{exp}\left(B\right)$ ? Study first the two examples

A_{1} = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}, \quad A_{2} = \begin{pmatrix} 2 & 1 \\ 0 & 1/2 \end{pmatrix}.

Both are elements of the Lie group $SL(2,\mathbb {R})$ . The matrix $A_{1}$ admits only one linearly independent eigenvector, namely $(1,0)^{T}$ . The matrix

B_{1} = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}

generates $A_{1}$ .

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Problem 14.2.13

Let $A$ be an $n \times n$ nonnormal matrix and $Q \in SL(n,\mathbb {C})$ . Can one find $Q$ such that $\widetilde A = QAQ^{-1}$ is normal?

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