Math Problems Vault

Problem 1.1.35 from Steeb W., Hardy Y. "Problems and Solutions in Introductory and Advanced Matrix Calculus" [StHa17]More problems in Section 1.1 "Chapter Problems"

Find the $2 \times 2$ matrices $F$ and $F'$ from the two equations

\begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} = \frac{1}{2}(F+iF'), \quad \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} = \frac{1}{2}(F-iF').

Find the anticommutator of $F$ and $F'$ , i.e. $[F,F']_{+} \equiv FF'+F'F$ .

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Problem 3.1.34 from Steeb W., Hardy Y. "Problems and Solutions in Introductory and Advanced Matrix Calculus" [StHa17]More problems in Section 3.1 "Chapter Problems"

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

(A \otimes I_{2}) \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 0 \\ 0 \\ 1 \end{pmatrix} = (I_{2} \otimes B) \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 0 \\ 0 \\ 1 \end{pmatrix}.

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Problem 4.1.78 from Steeb W., Hardy Y. "Problems and Solutions in Introductory and Advanced Matrix Calculus" [StHa17]More problems in Section 4.1 "Chapter Problems"

(i)(ii)

Let $M$ be the Minkowski space endowed with the standard coordinates $x_{0}, x_{1}, x_{2}, x_{3}$ and the metric tensor field

g = -dx_{0} \otimes dx_{0} + dx_{1} \otimes dx_{1} + dx_{2} \otimes dx_{2} + dx_{3} \otimes dx_{3}

and the quadratic form

q(x) = -(x_{0})^{2} + (x_{1})^{2} + (x_{2})^{2} + (x_{3})^{2}.

Let $\overrightarrow {H}(2)$ be the vector space of $2 \times 2$ hermitian matrices. Consider the map $\varphi : M \rightarrow \overrightarrow {H}(2)$

\varphi (x) = X = \begin{pmatrix} x_{0}+x_{3} & x_{1}-ix_{2} \\ x_{1}+ix_{2} & x_{0}-x_{3} \end{pmatrix}

which is an isomorphism.

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(i)

Find the determinant of $X$ .

(ii)

Find the Cayley transform of $X$ , i.e.

U = (X+iI_{2})^{-1}(X-iI_{2})

with the inverse

X = i(I_{2}+U)(I_{2}-U)^{-1}.

Problem 5.1.75 from Steeb W., Hardy Y. "Problems and Solutions in Introductory and Advanced Matrix Calculus" [StHa17]More problems in Section 5.1 "Chapter Problems"

Let $A, B$ be two nonzero $n \times n$ matrices over $\mathbb {C}$ . Let $A\overrightarrow {v}=\lambda \overrightarrow {v}$ be the eigenvalue equation for $A$ . Assume that $\left[A,B\right]=0_{n}$ . Then from $\left[A,B\right]\overrightarrow {v}=\overrightarrow {0}$ it follows that

\left[A,B\right]\overrightarrow {v} = (AB-BA)\overrightarrow {v} = A(B\overrightarrow {v})-B(A\overrightarrow {v}) = \overrightarrow {0}.

Therefore $A(B\overrightarrow {v})=\lambda (B\overrightarrow {v})$ . If $B\overrightarrow {v} \neq \overrightarrow {0}$ we find that $B\overrightarrow {v}$ is an eigenvector of $A$ with eigenvalue $\lambda$ . Apply it to $A=\sigma_{1} \otimes \sigma_{2}$ and $B=\sigma_{3} \otimes \sigma_{3}$ .

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Problem 5.1.76 from Steeb W., Hardy Y. "Problems and Solutions in Introductory and Advanced Matrix Calculus" [StHa17]More problems in Section 5.1 "Chapter Problems"

Let $A, B$ be two nonzero $n \times n$ matrices over $\mathbb {C}$ . Let $A\overrightarrow {v}=\lambda \overrightarrow {v}$ be the eigenvalue equation for $A$ . Assume that $\left[A,B\right]_{+}=0_{n}$ . Then from $\left[A,B\right]_{+}\overrightarrow {v}=\overrightarrow {0}$ it follows that

\left[A,B\right]_{+}\overrightarrow {v} = (AB+BA)\overrightarrow {v} = A(B\overrightarrow {v})+B(A\overrightarrow {v}) = \overrightarrow {0}.

Therefore $A(B\overrightarrow {v})=-\lambda (B\overrightarrow {v})$ . If $B\overrightarrow {v} \neq \overrightarrow {0}$ we have an eigenvalue equation with eigenvalue $-\lambda$ . Apply it to $A=\sigma_{1}$ and $B=\sigma_{2}$ , where $\left[\sigma_{1},\sigma_{2}\right]_{+}=0_{2}$ .

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Problem 5.1.78 from Steeb W., Hardy Y. "Problems and Solutions in Introductory and Advanced Matrix Calculus" [StHa17]More problems in Section 5.1 "Chapter Problems"

Consider the Pauli spin matrix $\sigma_{2}$ with the eigenvalues $+1, -1$ and the corresponding normalized eigenvectors

\frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ i \end{pmatrix}, \qquad \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ -i \end{pmatrix}.

Then $\sigma_{2} \otimes \sigma_{2}$ admits the eigenvalues $+1$ (twice) and $-1$ (twice). Show that there are product states as eigenvectors and two sets of entangled eigenvectors for $\sigma_{2} \otimes \sigma_{2}$ .

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Problem 5.1.85 from Steeb W., Hardy Y. "Problems and Solutions in Introductory and Advanced Matrix Calculus" [StHa17]More problems in Section 5.1 "Chapter Problems"

Consider the Hilbert space $\mathbb {C}^{n}$ . Let $A, B, C$ be $n \times n$ matrices acting in $\mathbb {C}^{n}$ . We consider the nonlinear eigenvalue problem

A\overrightarrow {v} = \lambda B\overrightarrow {v} + \lambda ^{2}C\overrightarrow {v}

where $\overrightarrow {v} \in \mathbb {C}^{n}$ and $\overrightarrow {v} \neq \overrightarrow {0}$ . Let $\sigma_{1}, \sigma_{2}, \sigma_{3}$ be the Pauli spin matrices. Find the solutions of the nonlinear eigenvalue problem

\sigma _{1}\overrightarrow {v} = \lambda \sigma _{2}\overrightarrow {v} + \lambda ^{2}\sigma _{3}\overrightarrow {v}

where $\overrightarrow {v} \in \mathbb {C}^{2}$ and $\overrightarrow {v} \neq \overrightarrow {0}$ .

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Problem 7.1.37 from Steeb W., Hardy Y. "Problems and Solutions in Introductory and Advanced Matrix Calculus" [StHa17]More problems in Section 7.1 "Chapter Problems"

(i)(ii)

The four Dirac matrices which play a role in the Dirac equation are given by

\gamma _1 = \begin{pmatrix} 0 & 0 & 0 & -i \\ 0 & 0 & -i & 0 \\ 0 & i & 0 & 0 \\ i & 0 & 0 & 0 \end{pmatrix}, \qquad \gamma _2 = \begin{pmatrix} 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \end{pmatrix},

\gamma _3 = \begin{pmatrix} 0 & 0 & -i & 0 \\ 0 & 0 & 0 & i \\ i & 0 & 0 & 0 \\ 0 & -i & 0 & 0 \end{pmatrix}, \qquad \gamma _4 = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix}.

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(i)

Show that the matrices can be written as Kronecker product of the Pauli spin matrices $\sigma_1$ , $\sigma_2$ , $\sigma_3$ and the $2 \times 2$ identity matrix $I_2$ .

(ii)

Calculate the anticommutators $[\gamma_4, \gamma_1]_+$ , $[\gamma_4, \gamma_2]_+$ , $[\gamma_4, \gamma_3]_+$ .

Problem 18.1.32 from Steeb W., Hardy Y. "Problems and Solutions in Introductory and Advanced Matrix Calculus" [StHa17]More problems in Section 18.1 "Chapter Problems"

Let $R \in \mathbb {C}^{m \times m}$ and $S \in \mathbb {C}^{n \times n}$ be nontrivial involutions. This means that $R = R^{-1} \neq \pm I_{m}$ and $S = S^{-1} \neq I_{n}$ . A matrix $A \in \mathbb {C}^{m \times n}$ is called $(R, S)$ -symmetric if $RAS = A$ . Consider the case $m = n = 2$ and the Pauli spin matrices

R = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad S = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}.

Find all $2 \times 2$ matrices $A$ over $\mathbb {C}$ such that $RAS = A$ .

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Problem 5.2.37 from Steeb W., Hardy Y. "Problems and Solutions in Introductory and Advanced Matrix Calculus" [StHa17]More problems in Section 5.2 "Supplementary Problems"

Let $\gamma \in \mathbb {R}$ and $\gamma^{2}<1$ . Consider the $2 \times 2$ matrix

K = \sigma _{1}-i\gamma \sigma _{3}.

Show that the eigenvalues of $K$ and $K^{*}$ are given by $\pm \sqrt{1-\gamma^{2}}$ . Find the normalized eigenvectors.

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Problem 18.2.43 from Steeb W., Hardy Y. "Problems and Solutions in Introductory and Advanced Matrix Calculus" [StHa17]More problems in Section 18.2 "Supplementary Problems"

(i)(ii)

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

\tau _{1} = i\sigma _{1}, \qquad \tau _{2} = i\sigma _{2}, \qquad \tau _{3} = i\sigma _{3}

and $\tau_{0} = I_{2}$ . Thus $\tau_{j}$ ( $j = 0, 1, 2, 3$ ) are elements of the Lie group $SU(2)$ .

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(i)

We define the operator $R$ as ( $k \geq 2$ )

R(\tau _{j_{1}} \otimes \tau _{j_{2}} \otimes \cdots \otimes \tau _{j_{k}}) := \frac{1}{2}(\tau _{j_{1}}\tau _{j_{2}}^{*} \otimes \tau _{j_{2}}\tau _{j_{3}}^{*} \otimes \cdots \otimes \tau _{j_{k}}\tau _{j_{1}}^{*}).

Find $R(\tau_{1} \otimes \tau_{2} \otimes \tau_{3})$ .

(ii)

Consider the unitary matrix (constructed from the four Bell states)

U = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 0 & 0 & -i \\ 0 & -i & -1 & 0 \\ 0 & -i & 1 & 0 \\ 1 & 0 & 0 & i \end{pmatrix} \Rightarrow U^{*} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 0 & 0 & 1 \\ 0 & i & i & 0 \\ 0 & -1 & 1 & 0 \\ i & 0 & 0 & -i \end{pmatrix}.

Show that $U(i\sigma_{1} \otimes i\sigma_{1})U$ , $U(i\sigma_{2} \otimes i\sigma_{2})U$ , $U(i\sigma_{3} \otimes i\sigma_{3})U$ are elements of the Lie group $SO(4)$ .

Pauli and Dirac matrices

References