Supplementary Problems

Given the standard basis $\left\{ \overrightarrow {e}_{1}, \overrightarrow {e}_{2}\right\}$ in $\mathbb {C}^{2}$. Is

a basis in $\mathbb {C}^{4}$?

Let $\overrightarrow {e}_{1}$, $\overrightarrow {e}_{2}$ be the standard basis in $\mathbb {C}^{2}$. The eight vectors

provides the standard basis in $\mathbb {R}^{8}$. Do the 8 vectors

form an orthonormal basis in $\mathbb {R}^{8}$?

Let the vectors $\overrightarrow {v}_{1}$, $\overrightarrow {v}_{2}$ form an orthonormal basis in $\mathbb {C}^{2}$. Then the four vectors

form an orthonormal basis in $\mathbb {C}^{4}$. Do the four vectors

form a basis in $\mathbb {C}^{4}$? Can the four vectors be written as a Kronecker product of two vectors in $\mathbb {C}^{2}$?

The three vectors

form an orthonormal basis in $\mathbb {R}^{3}$. Do the 9 vectors

form an orthonormal basis in $\mathbb {R}^{9}$? These vectors play a role for the $\pi$-mesons.

We know that if $A$ and $B$ normal matrices, then $A \otimes B$ is normal. Can we find two matrices $C$ and $D$ which are nonnormal but $C \otimes D$ is normal?

Consider the invertible nonnormal $2 \times 2$ matrix $M$

Is the matrix $M \otimes M^{-1}$ nonnormal?

Are the matrices linearly independent?

Are the matrices $A \otimes A$, $B \otimes B$, $C \otimes C$, $D \otimes D$ linearly independent?

Show that the $2 \times 2$ matrices $B$ and $D$ are similar. Show that the matrices $B \otimes D$ and $D \otimes B$ are similar.

Let $A$ be an $2 \times 2$ matrix over $\mathbb {C}$. Can $A \otimes A$ be written as

where $B_{1}$, $B_{2}$ are $2 \times 2$ matrices, $C_{1}$, $D_{2}$ are $1 \times 1$ matrices and $C_{2}$, $D_{1}$ are $3 \times 3$ matrices?

Let $A$, $B$, $C$, $D$ be $n \times n$ matrices. Is

Let $A$, $B$ be $2 \times 2$ matrices over $\mathbb {R}$. Can $A \otimes B$ be written as

Here $C$, $D$ are $2 \times 2$ matrices over $\mathbb {R}$ and $E$, $F$ are $3 \times 3$ matrices over $\mathbb {R}$.

Show that any $2 \times 2$ matrix $A \in SL(2,\mathbb {C})$ satisfies $A^{T}JA = J$, where $A^{T}$ is the transpose of $A$.

Let $A$ be a $2 \times 2$ matrix with $A \in SL(2,\mathbb {C})$. Show that

Find the conditions on the $2 \times 2$ matrices $A$ and $B$ such that

Let $A$, $B$ be $2 \times 2$ matrices. Find the conditions on $A$ and $B$ such that

Assume that $A \otimes B = B \otimes A$. Can we conclude that $AB = BA$? Prove or disprove.

Assume that $AB = BA$. Can be concluded that $A \otimes B = B \otimes A$?

Assume that

Find the coefficients $t_{jk\ell }$ such that

Assume that

Find the coefficients $t_{jk\ell }$ such that

Consider the vectors

in $\mathbb {C}^{3}$ and $\mathbb {C}^{2}$, respectively. Find the $6 \times 6$ permutation matrix $P$ such that $P(\overrightarrow {u} \otimes \overrightarrow {v}) = \overrightarrow {v} \otimes \overrightarrow {u}$.

Find all non-trivial solutions of the equation

From the equation we obtain the four conditions

A case study has to be done starting with $x_{1} = 0$ and $x_{1} \neq 0$.

Consider the standard basis $\overrightarrow {e}_{1}$, $\overrightarrow {e}_{2}$, $\overrightarrow {e}_{3}$ in $\mathbb {R}^{3}$. Find a $8 \times 8$ permutation matrix $T$ such that $T(\overrightarrow {e}_{1} \otimes \overrightarrow {e}_{2} \otimes \overrightarrow {e}_{3}) = \overrightarrow {e}_{3} \otimes \overrightarrow {e}_{2} \otimes \overrightarrow {e}_{1}$.

Show that any $2 \times 2$ matrix can be written as linear combination

Show that any $4 \times 4$ matrix can be written as linear combination

Show that any $2^{n} \times 2^{n}$ matrix can be written as linear combination

Consider the 256 sixteen times sixteen matrices

where $j_{0}, j_{1}, j_{2}, j_{3} \in \left\{ 0,1,2,3\right\}$. Let

Find $\psi^{T} T_{j_{0}j_{1}j_{2}j_{3}} \psi = t_{j_{0}j_{1}j_{2}j_{3}}$.

Find $X_{1}^{2}$, $X_{2}^{2}$, $X_{3}^{2}$, $X_{1}X_{2}$, $X_{1}X_{3}$, $X_{2}X_{3}$.

Find the commutators $[X_{1}, X_{2}]$, $[X_{2}, X_{3}]$, $[X_{3}, X_{2}]$.

Let $\overrightarrow {u}$, $\overrightarrow {v}$ be normalized column vectors in $\mathbb {C}^{n}$ with $\overrightarrow {u}^{*}\overrightarrow {v} = 0$. Show that $\overrightarrow {u} \otimes \overrightarrow {v} - \overrightarrow {v} \otimes \overrightarrow {u}$ cannot be written as

where $\overrightarrow {a}, \overrightarrow {b} \in \mathbb {C}^{n}$. This problem plays a role for entanglement.

Let $\overrightarrow {u}, \overrightarrow {v} \in \mathbb {C}^{n}$ and linearly independent. Can one find $\overrightarrow {x}, \overrightarrow {y} \in \mathbb {C}^{n}$ such that

Let $A$, $B$ be nonzero $n \times n$ matrices over $\mathbb {C}$. Assume that $A$ and $B$ are linearly independent. Show that $A \otimes B - B \otimes A$ cannot be written as $A \otimes B - B \otimes A = X \otimes Y$, where $X, Y$ are $n \times n$ matrices over $\mathbb {C}$.