Math Problems Vault

Problem 13.2.1

(i)(ii)

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

?

(i)

Consider the maps

\begin{align} f_{1}(A,B) & = A \otimes B \\ f_{2}(A,B) & = \operatorname {vec}(A)(\operatorname {vec}(B))^{T} \\ f_{3}(A,B) & = \operatorname {vec}(A)(\operatorname {vec}(B^{T}))^{T} \\ f_{4}(A,B) & = \operatorname {vec}(B)(\operatorname {vec}(A^{T}))^{T}. \end{align}

Is

\begin{align} \operatorname {tr}(f_{1}(A,B)) & = \operatorname {tr}(f_{2}(A,B)), \\ \operatorname {tr}(f_{2}(A,B)) & = \operatorname {tr}(f_{3}(A,B)), \\ \operatorname {tr}(f_{3}(A,B)) & = \operatorname {tr}(f_{4}(A,B)) \, ? \end{align}

(ii)

Show that

\operatorname {tr}(\operatorname {vec}(B)(\operatorname {vec}(A))^{T}) \equiv \operatorname {tr}(A^{T}B).

Problem 13.2.2

(i)(ii)

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

?

(i)

Find $\operatorname {vec}(A \otimes I_{2}+I_{2} \otimes A)$ and $\operatorname {vec}(A \otimes A)$ .

(ii)

Find the condition on $A$ such that $\operatorname {vec}(A \otimes A) = \operatorname {vec}(A \otimes I_{2}+I_{2} \otimes A)$ .

Problem 13.2.3

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

\operatorname {vec}(A \bullet B) = \operatorname {vec}(A) \bullet \operatorname {vec}(B)

where $\bullet$ denotes the Hadamard product.

?

Problem 13.2.4

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

\begin{align} X^{-1}\begin{pmatrix} 0 & -i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}X & = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end{pmatrix} \\ X^{-1}\begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -i \\ 0 & i & 0 \end{pmatrix}X & = \frac{1}{\sqrt{2}}\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix} \\ X^{-1}\begin{pmatrix} 0 & 0 & i \\ 0 & 0 & 0 \\ -i & 0 & 0 \end{pmatrix}X & = \frac{i}{\sqrt{2}}\begin{pmatrix} 0 & -1 & 0 \\ 1 & 0 & -1 \\ 0 & 1 & 0 \end{pmatrix}. \end{align}

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

\left( I_{3} \otimes \begin{pmatrix} 0 & -i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} - \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end{pmatrix}^{T} \otimes I_{3} \right)\operatorname {vec}(X) = \overrightarrow {0}

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

?

Problem 13.2.5

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

R := \sigma _{1} \otimes \sigma _{2} \equiv \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \otimes \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} \equiv \begin{pmatrix} 0 & 0 & 0 & -i \\ 0 & 0 & i & 0 \\ 0 & -i & 0 & 0 \\ i & 0 & 0 & 0 \end{pmatrix}.

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.

?

Problem 13.2.6

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

A = \begin{pmatrix} 1 & 2 & 3 \\ 3 & 1 & 2 \\ 2 & 3 & 1 \end{pmatrix}

where

S_{n} = \operatorname {vec}(I_{3}) = \begin{pmatrix} 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 \end{pmatrix}^{T}

and

S_{n}\operatorname {vec}(A) = \begin{pmatrix} 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 \end{pmatrix}\operatorname {vec}(A) = 3.

?

Problem 13.2.7

(i)(ii)

?

(i)

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

\operatorname {tr}(\operatorname {vec}(B)(\operatorname {vec}(A))^{T}) = \operatorname {tr}(A^{T}B)

applying $\operatorname {tr}(\operatorname {vec}(B)(\operatorname {vec}(A))^{T}) = \operatorname {tr}((\operatorname {vec}(I_{n}))^{T}(I \otimes A^{T}B)\operatorname {vec}(I))$ .

(ii)

Let $A$ be an $m \times n$ matrix, $B$ be a $p \times p$ matrix, $X$ be a $p \times n$ matrix and $C$ be an $n \times n$ matrix. The underlying field is $\mathbb {R}$ . Show that

\operatorname {tr}(AX^{T}BXC) \equiv (\operatorname {vec}(X))^{T}((CA) \otimes B^{T})\operatorname {vec}(X).