Linear matrix equations

Determine the commutant of the $n \times n$ Jordan matrix

In particular, determine the dimension of the commutant as a complex vector space.

What is the dimension of the commutant of the $2n \times 2n$ matrix

Show that the matrix equation $\sum_{i=1}^{k} x_{i}\mathbf{A}_{i} = \mathbf{B}$ (in unknowns $x_{1}, \ldots , x_{k}$) is equivalent to a linear system of the form $\mathbf{A}\mathbf{x} = \mathbf{b}$, where $\mathbf{x} = (x_{1}, \ldots , x_{k})'$ is a vector of unknowns.

Show that if $\mathbf{A}_{1}, \ldots , \mathbf{A}_{k}$, and $\mathbf{B}$ are symmetric, then the matrix equation $\sum_{i=1}^{k} x_{i}\mathbf{A}_{i} = \mathbf{B}$ (in unknowns $x_{1}, \ldots , x_{k}$) is equivalent to a linear system of the form $\mathbf{A}^{*}\mathbf{x} = \mathbf{b}^{*}$, where $\mathbf{b}^{*} = \operatorname {vech} \mathbf{B}$ and $\mathbf{x} = (x_{1}, \ldots , x_{k})'$ is a vector of unknowns.

$AX = O$

$AX = B$

$XA = B$

$AXC = B$

$A(X + C) = B$.

$\left[\begin{smallmatrix} 2 & 5 \\ 1 & 3\end{smallmatrix}\right] X =\left[\begin{smallmatrix} 2 & 1 \\ 1 & 1\end{smallmatrix}\right]$

$X\left[\begin{smallmatrix} 2 & 5 \\ 1 & 3\end{smallmatrix}\right] =\left[\begin{smallmatrix} 2 & 1 \\ 1 & 1\end{smallmatrix}\right]$

$\left[\begin{smallmatrix} 2 & 1 \\ 1 & 1\end{smallmatrix}\right] X =\left[\begin{smallmatrix} 10 \\ 17\end{smallmatrix}\right]$

$\left[\begin{smallmatrix} 1 & 2 & 0 \\ 2 & 5 & - 2 \\ 0 & - 2 & 5\end{smallmatrix}\right] X =\left[\begin{smallmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 1 & 0\end{smallmatrix}\right]$

$X\left[\begin{smallmatrix} 2 & 2 & - 1 \\ 2 & - 1 & 2 \\ - 1 & 2 & 2\end{smallmatrix}\right] =\left[\begin{smallmatrix} 5 & 5 & 2 \\ 5 & 8 & - 1\end{smallmatrix}\right]$

$X\left[\begin{smallmatrix} 1 & 1 \\ 1 & 1\end{smallmatrix}\right] =\left[\begin{smallmatrix} 1 & - 1 \\ 1 & - 1\end{smallmatrix}\right]$

$\left[\begin{smallmatrix} 1 & 1 \\ 0 & 1\end{smallmatrix}\right] X = X\left[\begin{smallmatrix} 1 & 1 \\ 0 & 1\end{smallmatrix}\right]$

$\mathbf{A}_{12} X = \mathbf{A}_{5}$. $\mathbf{A}_{12} = \left[\begin{smallmatrix} 1 & 1 \\ 1 & 1 \end{smallmatrix}\right], \mathbf{A}_{5} = \left[\begin{smallmatrix} 1 & -1 \\ 1 & -1 \end{smallmatrix}\right]$.

$X\mathbf{A}_{12} = \mathbf{A}_{12}$. $\mathbf{A}_{12} = \left[\begin{smallmatrix} 1 & 1 \\ 1 & 1 \end{smallmatrix}\right], \mathbf{A}_{12} = \left[\begin{smallmatrix} 1 & 1 \\ 1 & 1 \end{smallmatrix}\right]$.

$X^{ - 1} \mathbf{A}_{12} X = \mathbf{A}_{21}$. $\mathbf{A}_{12} = \left[\begin{smallmatrix} 1 & 1 \\ 1 & 1 \end{smallmatrix}\right], \mathbf{A}_{21} = \left[\begin{smallmatrix} 0 & 0 \\ 0 & 2 \end{smallmatrix}\right]$.

$\mathbf{A}_{111} X = \mathbf{A}_{228}$. $\mathbf{A}_{111} = \left[\begin{smallmatrix} 1 & 1 \\ 1 & 1 \\ 2 & 2 \end{smallmatrix}\right], \mathbf{A}_{228} = \left[\begin{smallmatrix} 1 & 2 & 4 \\ 1 & 2 & 4 \\ 2 & 4 & 8 \end{smallmatrix}\right]$.

$\mathbf{A}_{229} X = \mathbf{A}_{116}$. $\mathbf{A}_{229} = \left[\begin{smallmatrix} 2 & -1 & 0 \\ 0 & 2 & -1 \\ -1 & -1 & 1 \end{smallmatrix}\right], \mathbf{A}_{116} = \left[\begin{smallmatrix} 1 & 0 \\ 0 & 1 \\ -1 & -1 \end{smallmatrix}\right]$.

$\mathbf{A}_{203} X = \mathbf{c}_{53}$. $\mathbf{A}_{203} = \left[\begin{smallmatrix} 2 & 2 & -1 \\ 2 & -1 & 2 \\ -1 & 2 & 2 \end{smallmatrix}\right], \mathbf{c}_{53} = \left[\begin{smallmatrix} 7 \\ 1 \\ 1 \end{smallmatrix}\right]$.

$\mathbf{A}_{126} X = \mathbf{A}_{230}$. $\mathbf{A}_{126} = \left[\begin{smallmatrix} 3 & 2 \\ 3 & 1 \\ -2 & 0 \end{smallmatrix}\right], \mathbf{A}_{230} = \left[\begin{smallmatrix} -1 & 4 & 3 \\ -2 & 5 & 3 \\ 2 & -4 & -2 \end{smallmatrix}\right]$.

$\mathbf{A}_{113} X = \mathbf{A}_{213}$. $\mathbf{A}_{113} = \left[\begin{smallmatrix} 0 & 1 \\ 0 & 1 \\ 1 & 0 \end{smallmatrix}\right], \mathbf{A}_{213} = \left[\begin{smallmatrix} 1 & 1 & 1 \\ 2 & 2 & 2 \\ 3 & 3 & 3 \end{smallmatrix}\right]$.

$X\mathbf{A}_{227} = \mathbf{A}_{229}$. $\mathbf{A}_{227} = \left[\begin{smallmatrix} 1 & 1 & 1 \\ 1 & 2 & 2 \\ 2 & 3 & 4 \end{smallmatrix}\right], \mathbf{A}_{229} = \left[\begin{smallmatrix} 2 & -1 & 0 \\ 0 & 2 & -1 \\ -1 & -1 & 1 \end{smallmatrix}\right]$.

if $X$ is a matrix of size $n \times m$ and $A X-X B=0$, then $X=0$;

the equation $A X-X B=C$, where $X, C$ are matrices of size $n \times m$, has a solution, and moreover a unique one.

Consider the $3 \times 3$ permutation matrix

Find all $3 \times 3$ matrices $A$ such that $PAP^{T} = A$.

Consider the $4 \times 4$ permutation matrix

Find all $4 \times 4$ matrices $A$ such that $PAP^{T} = A$.

Find all $2 \times 2$ matrices $X$ such that

Impose the additional conditions such that $\operatorname {tr}(X) = 0$ and $\operatorname {det}\left(X\right) = +1$.

Let $A$ be an $m \times n$ matrix, $B$ an $p \times q$ matrix, and $C$ an $m \times q$ matrix. Let $X$ be an unknown $n \times p$ matrix. Show that the matrix equation $AXB = C$ is equivalent to the system of $qm$ equations in $np$ unknowns given by

that is, $\operatorname {vec}(AXB) = (B^{T} \otimes A) \operatorname {vec}(X)$.

Let $A$, $B$, $D$ be $n \times n$ matrices and $I_{n}$ the $n \times n$ identity matrix. Use the result from (i) to prove that $AX + XB = D$ can be written as

Consider the Pauli spin matrix $\sigma_1$ with eigenvalues $+1$ and $-1$ and corresponding normalized eigenvectors

Find all $2 \times 2$ matrices $A$ such that $[\sigma_1, A] = 0_2$. Find the eigenvalues and eigenvectors of $A$. Discuss.

Consider the counter-diagonal unit matrix

with eigenvalues $+1$ (twice) and $-1$ and the corresponding normalized eigenvectors

Find all $3 \times 3$ matrices $A$ such that $[J_3, A] = 0_3$. Find the eigenvalues and normalized eigenvectors of $A$ and compare with the eigenvectors of $J_3$. Discuss.

Consider the $4 \times 4$ matrix (counter-diagonal unit matrix)

with the eigenvalues $+1$ (twice) and $-1$ (twice) and the corresponding non-entangled eigenvectors

Find entangled eigenvectors. Find all $4 \times 4$ matrices $A$ such that $[A, J_4] \equiv AJ_4 - J_4A = 0_4$.

Let $J_n$ be the $n \times n$ counter unit matrix. Let $A$ be an $n \times n$ matrix with $[J_n, A] = 0_n$. Show that $[J_n \otimes J_n, A \otimes A] = 0_{n^{2}}$.

Let $AX+XB = C$, where $C$ is an $m \times n$ matrix over $\mathbb {R}$. What are the dimensions of $A$, $B$, and $X$?

Solve the equation

for the real-valued $2 \times 3$ matrix $X$.

Let $K$ be a given $n \times n$ nonnormal matrix over $\mathbb {C}$. We want to find all unitary $n \times n$ matrices such that $UKU^{*} = K^{*}$. Now we can write $KU^{*} = U^{*}K^{*}$ and therefore

where $0_{n}$ is the $n \times n$ zero matrix. This is a matrix equation (a special case of Sylvester) and using the vec-operator and Kronecker product we can cast the matrix equation into a vector equation. Write down this linear equation.

Apply it to the case $n=2$ with

Given the hermitian matrix $H$ we want to find the permutation matrices $P$ such that $HP = PH$. The equation can be written as $HP-PH = 0_{n}$. This is a special case of the Sylvester equation. Applying the vec-operation and the Kronecker product we can write the equation as

Consider $H = \sigma_{1}$. Find all $P$.

Given the permutation matrix $P$ we want to find the hermitian matrices $H$ such that $PH-HP = 0_{n}$. This is a special case of the Sylvester equation. Applying the vec-operation and the Kronecker product we can write the equation as

Consider $P = \sigma_{1}$. Find all $H$.

Let $n = 2$ and

Find all $4 \times 4$ matrices $R$ which satisfy the equation.

Let $n = 2$ and

Find all $T$ which satisfy the equation.

Find the solutions of the equation

Find the solutions of the equation

Given the $2 \times 2$ matrix $A$. Find all $2 \times 2$ matrices $X$ such that $AX = XA$. From $AX = XA$ we obtain

Thus we obtain the four equations

Show that the solution of the equation

is given by $x_{11} = x_{22}$, $x_{12} = x_{21}$.