Math Problems Vault

Problem 16.2.1

Consider the star product $A \star B$ of two $2 \times 2$ matrices $A$ and $B$ and the product

A \diamond B := \begin{pmatrix} b_{11} & 0 & 0 & b_{12} \\ b_{21} & 0 & 0 & b_{22} \\ 0 & a_{11} & a_{12} & 0 \\ 0 & a_{21} & a_{22} & 0 \end{pmatrix} .

Show that there is a $4 \times 4$ permutation matrix $P$ such that $P(A \star B) = A \diamond B$ .

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Problem 16.2.2

(i)(ii)(iii)

Consider the $2 \times 2$ matrices $A$ , $B$ over $\mathbb {C}$ . Let $A \star B$ be the star product.

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(i)

Let $A$ and $B$ be normal matrices. Show that $A \star B$ normal.

(ii)

Let $A$ and $B$ be unitary matrices. Show that $A \star B$ a unitary matrix.

(iii)

Let $A$ and $B$ be nilpotent matrices. Show that $A \star B$ a nilpotent matrix.

Problem 16.2.3

Let $A$ , $B$ be invertible $2 \times 2$ matrices. Let $A \star B$ be the star product. Show that

(A \star B)^{-1} = A^{-1} \star B^{-1} .

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Problem 16.2.4

(i)(ii)(iii)(iv)

Let $A$ be a $3 \times 3$ matrix and $B$ be a $2 \times 2$ matrix. We define

A \star B := \begin{pmatrix} b_{11} & 0 & 0 & 0 & b_{12} \\ 0 & a_{11} & a_{12} & a_{13} & 0 \\ 0 & a_{21} & a_{22} & a_{23} & 0 \\ 0 & a_{31} & a_{32} & a_{33} & 0 \\ b_{21} & 0 & 0 & 0 & b_{22} \end{pmatrix} .

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(i)

Find $\operatorname {det}\left(A \star B\right)$ and $\operatorname {tr}(A \star B)$ .

(ii)

Let $A$ , $B$ be normal matrices. Is $A \star B$ normal?

(iii)

Let $A$ , $B$ be unitary. Is $A \star B$ unitary?

(iv)

Let

A = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}, \qquad B = I_{3} .

Find $A \star B$ and calculate the eigenvalues and normalized eigenvectors of $A \star I_{3}$ .

Problem 16.2.5

Let $\Pi_{1}$ and $\Pi_{2}$ be $2 \times 2$ projection matrices. Is the $4 \times 4$ matrix $\Pi_{1} \star \Pi_{2}$ a projection matrix? Apply it to $\Pi_{1} \star \Pi_{1}$ where

\Pi _{1} = \frac{1}{2} \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} .

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Problem 16.2.6

(i)(ii)

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(i)

Let $U$ and $V$ be elements of the Lie group $SU(2)$ . Is $U \star V$ an element of the Lie group $SU(4)$ ?

(ii)

Let $X$ and $Y$ be elements of $SL(2, \mathbb {R})$ . Is $X \star Y$ an element of $SL(4, \mathbb {R})$ ?

Problem 16.2.7

Let $A$ , $B$ be normal $2 \times 2$ matrices with eigenvalues $\lambda_{1}, \lambda_{2}$ and $\mu_{1}, \mu_{2}$ , respectively. What can be said about the eigenvalues of $A \star B - B \star A$ ?

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Problem 16.2.8

(i)(ii)(iii)

Let $A$ , $B$ be $2 \times 2$ matrices.

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(i)

What are the conditions on $A$ and $B$ such that $\widehat{A \star B} = A \oplus B$ ?

(ii)

What are the conditions on $A$ and $B$ such that $\widehat{A \star B} = A \otimes B$ ?

(iii)

What are the conditions on $A$ and $B$ such that $A \star B = A \oplus B$ ?