Math Problems Vault

Problem 16.1.1

(i)(ii)

Consider the unitary and hermitian $2 \times 2$ matrix

U = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} .

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(i)

Find $U \star U$ . Is $U \star U$ unitary? Is $U \star U$ hermitian?

(ii)

Find $\widehat{U \star U}$ . Is $\widehat{U \star U}$ unitary? Is $\widehat{U \star U}$ hermitian?

Problem 16.1.2

(i)(ii)

Consider the nonnormal $2 \times 2$ matrices

A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \qquad B = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} = A^{T}

with the eigenvalues $0$ $(2\times )$ .

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(i)

Find $A \star B$ and $B \star A$ and the eigenvalues.

(ii)

Is $A \star B$ nonnormal? Is $B \star A$ nonnormal?

Problem 16.1.3

(i)(ii)

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(i)

What can be said about the trace of $A \star B$ ? What can be said about the determinant of $A \star B$ ?

(ii)

What can be said about the trace of $\widehat{A \star B}$ ? What can be said about the determinant of $\widehat{A \star B}$ ?

Problem 16.1.4

Let $A$ , $B$ be $2 \times 2$ matrices. Can one find a $4 \times 4$ permutation matrix $P$ such that $P(A \star B)P^{T} = B \oplus A$ ? Here $\oplus$ denotes the direct sum.

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Problem 16.1.5

Let $A$ , $B$ be $2 \times 2$ matrices. Let $A \star B$ be the star product. Show that one can find a $4 \times 4$ permutation matrix $P$ such that

P(A \star B)P^{T} = \begin{pmatrix} a_{11} & a_{12} & 0 & 0 \\ a_{21} & a_{22} & 0 & 0 \\ 0 & 0 & b_{11} & b_{12} \\ 0 & 0 & b_{21} & b_{22} \end{pmatrix} .

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Problem 16.1.6

The $2 \times 2$ matrices

I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \qquad J = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}

form a group under matrix multiplication. Do the four $4 \times 4$ matrices $I \star I$ , $I \star J$ , $J \star I$ , $J \star J$ form a group under matrix multiplication?

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Problem 16.1.7

Given the eigenvalues and eigenvectors of the $2 \times 2$ matrices $A$ and $B$ . What can be said about the eigenvalues and eigenvectors of $A \star B$ ?

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Problem 16.1.8

The $2 \times 2$ matrix

A(\alpha ) = \begin{pmatrix} \cos (\alpha ) & -\sin (\alpha ) \\ \sin (\alpha ) & \cos (\alpha ) \end{pmatrix}

admits the eigenvalues $\lambda_{+}(\alpha ) = e^{i\alpha }$ and $\lambda_{-}(\alpha ) = e^{-i\alpha }$ with the corresponding normalized eigenvectors

\frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ -i \end{pmatrix}, \qquad \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ i \end{pmatrix} .

Let

A(\alpha ) \star A(\alpha ) = \begin{pmatrix} \cos (\alpha ) & 0 & 0 & -\sin (\alpha ) \\ 0 & \cos (\alpha ) & -\sin (\alpha ) & 0 \\ 0 & \sin (\alpha ) & \cos (\alpha ) & 0 \\ \sin (\alpha ) & 0 & 0 & \cos (\alpha ) \end{pmatrix} .

Find the eigenvalues and normalized eigenvectors of $A(\alpha ) \star A(\alpha )$ .

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Problem 16.1.9

Let $A$ , $B$ be two $2 \times 2$ matrices and $A \star B$ be the star product. Can one find a $4 \times 4$ permutation matrix $P$ such that

P(A \star B) = \widehat{A \star B} \, ?

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Problem 16.1.10

Let $A$ , $B$ be $2 \times 2$ matrices. Find $A \star B - B \star A$ .

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