Star Product

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

Consider the nonnormal $2 \times 2$ matrices

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

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.

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

The $2 \times 2$ matrices

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?

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

The $2 \times 2$ matrix

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

Let

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

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

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

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

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

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

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

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

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

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 $A \star B - B \star A$?

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