vec Operator

Consider the $2 \times 3$ matrix

Let $B = A^{T}$. Thus $B$ is a $3 \times 2$ matrix. Find the $6 \times 6$ permutation matrix $P$ such that

Let $A$ be an $m \times n$ matrix over $\mathbb {C}$ and $B$ be a $s \times t$ matrix over $\mathbb {C}$. Find the permutation matrix $P$ such that

Let $\overrightarrow {v}$ be a column vector in $\mathbb {R}^{n}$. Show that $\operatorname {vec}(\overrightarrow {v}\overrightarrow {v}^{T}) = \overrightarrow {v} \otimes \overrightarrow {v}$.

Let $A$ be an $m \times n$ matrix over $\mathbb {C}$. Using

and

Show that $\operatorname {vec}_{m \times n}^{-1}(\operatorname {vec}_{m \times n}(A)) = A$.

Let $A$ be an $m \times n$ matrix over $\mathbb {C}$ and $B$ be an $s \times t$ matrix over $\mathbb {C}$. For the rearrangement operator $R$ defined by

and linear extension. Find

Consider Lyapunov $FX+XF^{T} = C$ and

Find $X$.

Consider the matrices

Find the solution of the equation $AX+XB = C$, where $X$ is the $3 \times 2$ matrix

Let $A$, $B$, $X$ be $n \times n$ matrices, where the matrices $A$, $B$ are given. Assume that $AX = B$. Find $X$ using the vec-operator and the Kronecker product. Then consider the case that $A$ is invertible.

Let $A$, $B$, $C$, $X$ be $n \times n$ matrices, where the matrices $A$, $B$, $C$ are given. Assume that $AXB = C$. Find $X$ using the vec operator and the Kronecker product. Then consider the case that $A$ and $B$ are invertible.

Let $A$, $B$, $C$, $X$, $Y$ be $n \times n$ matrices, where the matrices $A$, $B$, $C$ are given. Assume that $AX+YB = C$. Apply the vec operator to this equation and then express the left-hand side using the Kronecker product.

Let $U$ be an $n \times n$ unitary matrix. Show that the column vector $\operatorname {vec}(U)$ cannot be written as a Kronecker product of two vectors in $\mathbb {C}^{n}$.

Let $A$ be an arbitrary $n \times n$ matrix. Let $U$ be an $n \times n$ unitary matrix $U^{-1} = U^{*}$. Can we conclude that

Consider the Frobenius norm. Let $M$ be an $ms \times nt$ matrix over $\mathbb {C}$. Find the $m \times n$ matrices $A$ over $\mathbb {C}$ and the $s \times t$ matrices $B$ over $\mathbb {C}$ which minimize

Apply the method for $m=2, n=1, s=2, t=2$ to the $4 \times 2$ matrix

Let $P$ be an $n \times n$ permutation matrix and $H$ be an $n \times n$ hermitian matrix.

Consider the Pauli spin matrices $\sigma_{1}$, $\sigma_{3}$. The unitary and hermitian normal matrices $\sigma_{1}$ and $\sigma_{3}$ have the same eigenvalues, namely $+1$ and $-1$. Find all invertible $2 \times 2$ matrices $X$ such that

Proceed as follows: Multiply the equation with $X$ and after rearrangements we have

This is a special case of the Sylvester equation. It can be written as

Solve this linear equation to find the matrix $X$, but one still has to take into account that the matrix $X$ is invertible.

Let $\mathbf{A} \in \mathbb {C}^{n^{m}} \equiv \bigotimes^{m}\mathbb {C}^{n}$ and $\overrightarrow {x} \in \mathbb {C}^{n}$. Define the product

Let $A$ be an $n \times n$ matrix over $\mathbb {C}$. Find $\operatorname {vec}(A^{T})\overrightarrow {x}^{1}$. What is the relationship between this product and the matrix product $A\overrightarrow {x}$?

Let $\mathbf{A} \in \mathbb {C}^{n^{m}} \equiv \bigotimes^{m}\mathbb {C}^{n}$. If $\lambda \in \mathbb {C}$ and $\overrightarrow {x} \in \mathbb {C}^{n}$, $\overrightarrow {x} \neq \overrightarrow {0}$, satisfies $\mathbf{A}\overrightarrow {x}^{m-1} = \lambda \overrightarrow {x}$ then $(\lambda , \overrightarrow {x})$ is said to be an eigenpair of $\mathbf{A}$, where $\lambda$ is the eigenvalue of $\mathbf{A}$ and $\overrightarrow {x}$ is a corresponding eigenvector. Two eigenpairs $(\lambda , \overrightarrow {x})$ and $(\lambda ', \overrightarrow {x}')$ are equivalent when $t^{m-2}\lambda = \lambda '$ and $t\overrightarrow {x} = \overrightarrow {x}'$ for some non-zero complex number $t$.

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

Let $A$ be a $2 \times 2$ matrix.

Let $A$, $B$ be $n \times n$ matrices. Show that

where $\bullet$ denotes the Hadamard product.

Find an invertible $3 \times 3$ matrix $X$ which satisfies the following three matrix equations simultaneously

Cast the equations into a Sylvester equation and then solve the equations. The first equation can be written as

where $\overrightarrow {0}$ is the zero column vector.

Consider the Pauli spin matrices $\sigma_{1}$, $\sigma_{2}$ and

Apply the vec-operator on $R$, i.e. $\operatorname {vec}(R)$. Calculate the discrete Fourier transform of the vector in $\mathbb {C}^{4}$. Then apply $\operatorname {vec}^{-1}$ to this vector. Compare this $4 \times 4$ matrix with the $4 \times 4$ matrix $R$. Discuss.

Let $A$ be an $n \times n$ matrix over $\mathbb {C}$. Find a matrix $S_{n}$ such that $S_{n}\operatorname {vec}(A) = \operatorname {tr}(A)$. Prove the equivalence. Illustrate the answer with

where

and