Kronecker Product

Consider the Pauli spin matrices

Find $\sigma_{1} \otimes \sigma_{3}$ and $\sigma_{3} \otimes \sigma_{1}$. Discuss. Find $\operatorname {tr}(\sigma_{1} \otimes \sigma_{3})$ and $\operatorname {tr}(\sigma_{3} \otimes \sigma_{1})$.

Let

Thus $\left\{ \overrightarrow {x}_{1}, \overrightarrow {x}_{2}\right\}$ forms an orthonormal basis in $\mathbb {R}^{2}$. Calculate

and interpret the result.

Given the orthonormal basis

in the vector space $\mathbb {C}^{2}$. Use this orthonormal basis to find an orthonormal basis in $\mathbb {C}^{4}$.

Consider the two normalized vectors

in the Hilbert space $\mathbb {R}^{4}$. Show that the vectors obtained by applying the matrix $(I_{2} \otimes \sigma_{1})$ together with the original ones form an orthonormal basis in $\mathbb {R}^{4}$.

Let $A$ be an $m \times m$ matrix and $B$ be an $n \times n$ matrix. The underlying field is $\mathbb {C}$. Let $I_{m}$, $I_{n}$ be the $m \times m$ and $n \times n$ unit matrix, respectively.

Let $A$, $B$ be arbitrary $n \times n$ matrices over $\mathbb {C}$. Show that

Let $A$ and $B$ be arbitrary $n \times n$ matrices over $\mathbb {C}$. Is $e^{A \otimes B} = e^{A} \otimes e^{B}$?

The underlying field is $\mathbb {C}$. Let $A$ be an $m \times m$ matrix with eigenvalue $\lambda$ and corresponding normalized eigenvector $\overrightarrow {v}$. Let $B$ be an $n \times n$ matrix with eigenvalue $\mu$ and corresponding normalized eigenvector $\overrightarrow {u}$. Let $\epsilon_{1}$, $\epsilon_{2}$ and $\epsilon_{3}$ be real parameters. Find an eigenvalue and the corresponding normalized eigenvectors of the matrix

Let $A_{j}$ ($j = 1, \ldots , k$) be matrices of size $m_{j} \times n_{j}$. We introduce the notation

Consider the elementary matrices

A square matrix is called a stochastic matrix if each entry is nonnegative and the sum of the entries in each row is 1. Let $A$, $B$ be $n \times n$ stochastic matrices. Is $A \otimes B$ a stochastic matrix?

Let $X$ be an $m \times m$ and $Y$ be an $n \times n$ matrix. The direct sum is the $(m+n) \times (m+n)$ matrix

Let $A$ be an $n \times n$ matrix, $B$ be an $m \times m$ matrix and $C$ be an $p \times p$ matrix. Then we have the identity

Is $A \otimes (B \oplus C) = (A \otimes B) \oplus (A \otimes C)$ true?

Let $A$, $B$ be $n \times n$ matrices. Then the direct sum $A \oplus B$ is the $2n \times 2n$ matrix

Let $A$, $B$ be $2 \times 2$ matrices, $C$ a $3 \times 3$ matrix and $D$ a $1 \times 1$ matrix. Find the condition on these matrices such that $A \otimes B = C \oplus D$, where $\oplus$ denotes the direct sum. We assume that $D$ is nonzero.

Let $A$ be an $2 \times 2$ matrix over $\mathbb {C}$. Solve the equation $A \oplus A = A \otimes A$.

With each $m \times n$ matrix $Y$ we associate the column vector $\operatorname {vec}(Y)$ of length $m \times n$ defined by

Let $A$, $B$, $C$, $D$ be symmetric $n \times n$ matrices over $\mathbb {R}$. Assume that these matrices commute with each other. Consider the $4n \times 4n$ matrix

Let $\overrightarrow {x}, \overrightarrow {y}, \overrightarrow {z} \in \mathbb {R}^{n}$. We define a wedge product

Show that $(\overrightarrow {x} \wedge \overrightarrow {y}) \wedge \overrightarrow {z} + (\overrightarrow {z} \wedge \overrightarrow {x}) \wedge \overrightarrow {y} + (\overrightarrow {y} \wedge \overrightarrow {z}) \wedge \overrightarrow {x} = \overrightarrow {0}$.

Consider the vectors $\overrightarrow {v}$ and $\overrightarrow {w}$ in $\mathbb {C}^{2}$

Let $A$, $B$ be skew-hermitian matrices, i.e.$A^{*} = -A$ and $B^{*} = -B$. Is $A \otimes B$ skew-hermitian?

Let $A$, $B$ be $n \times n$ normal matrices over $\mathbb {C}$, i.e.$AA^{*} = A^{*}A$ and $BB^{*} = B^{*}B$. Show that $A \otimes B$ is a normal matrix.

Let $\overrightarrow {z} = \begin{pmatrix} z_{0} & z_{1} & \ldots & z_{N} \end{pmatrix}^{T}$ be a (column) vector in $\mathbb {C}^{N+1}$ and assume that $\overrightarrow {z}$ is nonzero. Then

i.e.$\overrightarrow {z}^{*}$ is the transpose and conjugate complex of $\overrightarrow {z}$. Consider the $(N+1) \times (N+1)$ matrix

Show that $\Pi^{*} = \Pi$, $\Pi^{2} = \Pi$. This means that $\Pi$ is a projection matrix.

Let $A$, $B$ be $n \times n$ tridiagonal matrices with $n \geq 3$. Is $A \otimes B$ a tridiagonal matrix?

Consider the $2 \times 2$ matrix $A$ over $\mathbb {C}$ with $\operatorname {det}\left(A\right) = 1$.

Let $A$ be an $n \times n$ hermitian matrix with $A^{2} = I_{n}$. Let

Show that $\Pi_{1}$ and $\Pi_{2}$ are projection matrices. Calculate the product $\Pi_{1} \Pi_{2}$.

Is the Kronecker product of two circulant matrices again a circulant matrix? Let $A$, $B$ be two $n \times n$ circulant matrices. Is $A \otimes B = B \otimes A$?

A Hankel matrix is a matrix where for each antidiagonal the element is the same. For example, Hankel matrices are

Given two square Hankel matrices. Is the Kronecker product again a Hankel matrix?

Let $A$ be an invertible $n \times n$ matrix. Then the matrix $(A^{-1} \otimes I_{n})(I_{n} \otimes A)$ also has an inverse. Find the inverse.

Let $A$, $B$ be symmetric matrices over $\mathbb {R}$.

Let $A$, $B$ be $2 \times 2$ matrices over $\mathbb {R}$. Find the conditions on $A$ and $B$ such that

Let $\sigma_{1}$, $\sigma_{2}$, $\sigma_{3}$ be the Pauli spin matrices.

Let $A$ be an $n \times n$ matrix. Show that the $2n \times 2n$ matrix

can be expressed using the Kronecker product, the $n \times n$ identity matrix $I_{n}$, the matrix $A$ and the $2 \times 2$ matrices

Let $E_{jk}$ ($j, k = 1, \ldots , n$) be the $n \times n$ matrices with 1 at entry $(j, k)$ and 0 otherwise. Let $n = 2$. Find the $4 \times 4$ matrices

and

Are the matrices invertible?

Let $A$, $B$, $C$ be $2 \times 2$ matrices. Can we find $8 \times 8$ permutation matrices $P$ and $Q$ such that $P(A \otimes B \otimes C)Q = C \otimes B \otimes A$?

Let $P_{1}$ be an $m \times m$ permutation matrix and $P_{2}$ an $n \times n$ permutation matrix. Then $P_{1} \otimes P_{2}$ is an $mn \times mn$ permutation matrix. Let $K_{1}$ and $K_{2}$ be $m \times m$ and $n \times n$ matrices, respectively such that $P_{1} = e^{K_{1}}$ and $P_{2} = e^{K_{2}}$. Find $K$ such that $P_{1} \otimes P_{2} = e^{K}$, where $K$ is an $mn \times mn$ matrix.

Let $A$ and $B$ be two complex valued matrices, not necessarily of the same size. If $B \neq 0$ then the operation $\oslash$ given by

is well defined, where $\otimes$ is the Kronecker product. The operation $\oslash$ is called a right Kronecker quotient. Show that the nearest Kronecker product

defines a Kronecker quotient, where $\left\| \cdot \right\|$ denotes the Frobenius norm. Show that

where $\operatorname {tr}_{2}$ denotes the partial trace over matrices with the same size as $B^{*}B$, and $I$ is an appropriate identity matrix. Are there any other Kronecker quotients?

Let $d \geq 2$. Consider the vector space $V = \mathbb {C}^{d}$. Let $V \times V$ be the Cartesian product of $V$ with itself. Let $\overrightarrow {u}$, $\overrightarrow {v}$, $\overrightarrow {w}$ be elements of $V$. The Kronecker product (tensor product) is defined by the relations

Let $V \otimes V \equiv V^{\otimes^{2}} = \left\langle \overrightarrow {u} \otimes \overrightarrow {v} : \overrightarrow {u}, \overrightarrow {v} \in V\right\rangle$, where the span is taken in $\mathbb {C}^{d^{2}}$. Let $\left\{ \overrightarrow {e}_{j}\right\}_{j=1}^{d}$ be a basis in the vector space $V$. Then $\left\{ \overrightarrow {e}_{j} \otimes \overrightarrow {e}_{k}\right\}$ ($j, k = 1, \ldots , d$) is a basis in the vector space $V \otimes V$ with dimension $d^{2}$. We define the $k$-th tensor power of the vector space $V$ denoted by $V^{\otimes^{k}}$ given by the elements

Thus the dimension of the vector space $V^{\otimes^{k}}$ is $d^{k}$. Let $k \leq d$. Then the wedge product (also called Grassmann product, exterior product) is defined by

where $S_{k}$ is the permutation group and $\sigma$ is its alternating representation, i.e.$\sigma (\pi ) = +1$ for even permutations and $\sigma (\pi ) = -1$ for odd permutations. The complex vector space spanned by elements of the form

is denoted by $\wedge^{k} V$ and has dimension $\binom {d}{k}$. Give a computer algebra implementation with SymbolicC++ of the wedge product. Apply it to the case $d = 3$ with the basis

Calculate $\overrightarrow {v}_{1} \wedge \overrightarrow {v}_{2}$, $\overrightarrow {v}_{2} \wedge \overrightarrow {v}_{3}$, $\overrightarrow {v}_{3} \wedge \overrightarrow {v}_{1}$, $\overrightarrow {v}_{1} \wedge \overrightarrow {v}_{2} \wedge \overrightarrow {v}_{3}$.