Math Problems Vault

Problem 17.1.1

(i)(ii)(iii)

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(i)

Let $\phi_{1}, \phi_{2} \in \mathbb {R}$ . Are the $2 \times 2$ matrices

U_{1} = \begin{pmatrix} e^{i\phi _{1}} & 0 \\ 0 & e^{i\phi _{2}} \end{pmatrix}, \qquad U_{2} = \begin{pmatrix} 0 & e^{i\phi _{1}} \\ e^{i\phi _{2}} & 0 \end{pmatrix}

unitary?

(ii)

Are the $2 \times 2$ matrices

V_{1} = \begin{pmatrix} 1/2 & \sqrt{3}/2 \\ \sqrt{3}/2 & -1/2 \end{pmatrix}, \qquad V_{2} = \begin{pmatrix} 1/2 & -\sqrt{3}/2 \\ -\sqrt{3}/2 & -1/2 \end{pmatrix}

unitary?

(iii)

Let $\theta , \phi \in \mathbb {C}$ . Is the $2 \times 2$ matrix

U(\theta ,\phi ) = \begin{pmatrix} \cos (\theta ) & e^{i\phi }\sin (\theta ) \\ e^{-i\phi }\sin (\theta ) & -\cos (\theta ) \end{pmatrix}

unitary? If so find the inverse.

Problem 17.1.2

(i)(ii)(iii)

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(i)

Can one find a $2 \times 2$ unitary matrix such that

U \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} U^{-1} = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix} ?

(ii)

Find a $2 \times 2$ unitary matrix $V$ such that

V^{*} \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} V = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}.

(iii)

Can one find a $2 \times 2$ unitary matrix $W$ such that

W \begin{pmatrix} 1 & 0 \\ 1 & 0 \end{pmatrix} W^{-1} = \begin{pmatrix} 0 & 1 \\ 0 & 1 \end{pmatrix} ?

Problem 17.1.3

Let $0 \leq r_{jk} \leq 1$ and $\phi_{jk} \in \mathbb {R}$ ( $j,k = 1,2$ ). Find the conditions on $r_{jk}, \phi_{jk}$ such that the $2 \times 2$ matrix

U(r_{jk}, \phi _{jk}) = \begin{pmatrix} r_{11}e^{i\phi _{11}} & r_{12}e^{i\phi _{12}} \\ r_{21}e^{i\phi _{21}} & r_{22}e^{i\phi _{22}} \end{pmatrix}

is unitary. Then simplify to the special case $\phi_{jk} = 0$ for $j,k = 1,2$ .

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Problem 17.1.4

(i)(ii)(iii)

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(i)

Is the $3 \times 3$ matrix

U_{1} = \begin{pmatrix} -1/\sqrt{2} & 0 & 1/\sqrt{2} \\ -i/\sqrt{2} & 0 & -i/\sqrt{2} \\ 0 & 1 & 0 \end{pmatrix}

unitary?

(ii)

Is the $3 \times 3$ matrix

U_{2} = \begin{pmatrix} 1/\sqrt{3} & 1/\sqrt{3} & 1/\sqrt{3} \\ 1/\sqrt{6} & 1/\sqrt{6} & -\sqrt{2}/\sqrt{3} \\ 1/\sqrt{2} & -1/\sqrt{2} & 0 \end{pmatrix}

a unitary matrix?

(iii)

Consider the $3 \times 3$ unitary matrix

U = \begin{pmatrix} 1/\sqrt{2} & 0 & -1/\sqrt{2} \\ 1/\sqrt{2} & 0 & 1/\sqrt{2} \\ 0 & -1 & 0 \end{pmatrix}

and the normalized vector in $\mathbb {C}^{3}$

\overrightarrow {v} = \frac{1}{\sqrt{3}}\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}.

Find the vectors $U\overrightarrow {v}$ and $U^{2}\overrightarrow {v}$ .

Problem 17.1.5

(i)(ii)

Consider the orthonormal basis in $\mathbb {C}^{3}$

\overrightarrow {v}_{1} = \frac{1}{\sqrt{2}}\begin{pmatrix} i \\ 0 \\ i \end{pmatrix}, \qquad \overrightarrow {v}_{2} = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \qquad \overrightarrow {v}_{3} = \frac{1}{\sqrt{2}}\begin{pmatrix} i \\ 0 \\ -i \end{pmatrix}.

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(i)

Is $U = \begin{pmatrix} \overrightarrow {v}_{1} & \overrightarrow {v}_{2} & \overrightarrow {v}_{3} \end{pmatrix}$ a unitary matrix?

(ii)

Let $\phi_{1}, \phi_{2}, \phi_{3} \in \mathbb {R}$ . Is $V(\phi_{1},\phi_{2},\phi_{3}) = e^{i\phi_{1}}\overrightarrow {v}_{1}\overrightarrow {v}_{1}^{*} + e^{i\phi_{2}}\overrightarrow {v}_{2}\overrightarrow {v}_{2}^{*} + e^{i\phi_{3}}\overrightarrow {v}_{3}\overrightarrow {v}_{3}^{*}$ a unitary matrix?

Problem 17.1.6

(i)(ii)

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(i)

Let $A$ , $B$ be $n \times n$ matrices over $\mathbb {R}$ . Show that one can find a $2n \times 2n$ unitary matrix $U$ such that

U \begin{pmatrix} A & B \\ -B & A \end{pmatrix} U^{*} = \begin{pmatrix} A+iB & 0_{n} \\ 0_{n} & A-iB \end{pmatrix}.

Here $0_{n}$ denotes the $n \times n$ zero matrix.

(ii)

Use the result from (i) to show that

\det \begin{pmatrix} A & B \\ -B & A \end{pmatrix} = \operatorname {det}\left(A+iB\right)\, \overline{\operatorname {det}\left(A+iB\right)} \geq 0.

Problem 17.1.7

(i)(ii)(iii)

Let $\overrightarrow {v}$ be a column vector in $\mathbb {C}^{n}$ with $\overrightarrow {v}^{*}\overrightarrow {v} = 1$ , i.e. the vector is normalized. Consider the matrix $U = I_{n} - 2\overrightarrow {v}\overrightarrow {v}^{*}$ .

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(i)

Show that $U$ is hermitian.

(ii)

Show that $U$ is unitary.

(iii)

Let $n = 2$ and $\overrightarrow {v}^{*} = \frac{1}{\sqrt{2}}(1 \; 1)$ . Find $U$ .

Problem 17.1.8

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(i)

Let $U$ be a unitary matrix with $U^{2} = I_{n}$ . Can we conclude that $U$ is hermitian?

(ii)

Let $\alpha \in \mathbb {R}$ . Calculate $\mathrm{exp}\left(\alpha U\right)$ .

(iii)

Let $\alpha \in \mathbb {R}$ . Consider the $2 \times 2$ unitary matrix

U_{1} = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}.

Calculate $\mathrm{exp}\left(i\alpha U_{1}\right)$ .

(iv)

Let $\alpha \in \mathbb {R}$ . Consider the $2 \times 2$ unitary matrix

U_{2} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}.

Calculate $\mathrm{exp}\left(\alpha U_{2}\right)$ .

(v)

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

\sigma _{\phi } := \cos (\phi )\sigma _{1}+\sin (\phi )\sigma _{2} = \begin{pmatrix} 0 & \cos (\phi )-i\sin (\phi ) \\ \cos (\phi )+i\sin (\phi ) & 0 \end{pmatrix}.

Calculate $\mathrm{exp}\left(-i\theta \sigma_{\phi }/2\right)$ .

Problem 17.1.9

Consider the two $2 \times 2$ unitary matrices

U_{1} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \qquad U_{2} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}.

Can one find a unitary $2 \times 2$ matrix $V$ such that $U_{1} = VU_{2}V^{*}$ ?

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Problem 17.1.10

(i)(ii)

Let $U$ be an $n \times n$ unitary matrix.

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(i)

Is $U+U^{*}$ invertible?

(ii)

Is $U+U^{*}$ hermitian?

Problem 17.1.11

Find the condition on the $n \times n$ matrix $A$ over $\mathbb {C}$ such that $I_{n}+A$ is a unitary matrix.

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Problem 17.1.12

Let $z_{1}, z_{2}, w_{1}, w_{2} \in \mathbb {C}$ . Consider the $2 \times 2$ matrices

U = \begin{pmatrix} 0 & z_{1} \\ z_{2} & 0 \end{pmatrix}, \qquad V = \begin{pmatrix} 0 & w_{1} \\ w_{2} & 0 \end{pmatrix}

where $z_{1}\overline{z_{1}} = 1$ , $z_{2}\overline{z_{2}} = 1$ , $w_{1}\overline{w_{1}} = 1$ , $w_{2}\overline{w_{2}} = 1$ . This means the matrices $U$ , $V$ are unitary. Find the condition on $z_{1}, z_{2}, w_{1}, w_{2}$ such that the commutator $[U,V]$ is again a unitary matrix.

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Problem 17.1.13

Find a unitary $3 \times 3$ matrix $U$ with $\operatorname {det}\left(U\right) = -1$ such that

U \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}.

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Problem 17.1.14

Show that the two matrices

A = \begin{pmatrix} e^{i\theta } & 0 \\ 0 & e^{-i\theta } \end{pmatrix}, \qquad B = \begin{pmatrix} \cos (\theta ) & \sin (\theta ) \\ -\sin (\theta ) & \cos (\theta ) \end{pmatrix}

are conjugate in the Lie group $SU(2)$ .

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Problem 17.1.15

Let $U$ be an $n \times n$ unitary matrix. Let $V$ be an $n \times n$ unitary matrix such that $V^{-1}UV = D$ is a diagonal matrix $D$ . Is $V^{-1}U^{*}V$ a diagonal matrix?

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Problem 17.1.16

Let $n \geq 2$ and even. Let $U$ be a unitary antisymmetric $n \times n$ matrix. Show that there exists a unitary matrix $V$ such that

V^{T}UV = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \oplus \cdots \oplus \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}

where $\oplus$ denotes the direct sum.

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Problem 17.1.17

Let $U$ be a unitary and symmetric matrix. Show that there exists a unitary and symmetric matrix $V$ such that $U = V^{2}$ .

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Problem 17.1.18

Let $U$ be an $n \times n$ unitary matrix. Show that $\left|\operatorname {det}\left(U\right)\right| = 1$ .

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Problem 17.1.19

Let $A$ be an $n \times n$ matrix over $\mathbb {R}$ . Show that if $\lambda$ is an eigenvalue of $A$ , then $\overline{\lambda }$ is also an eigenvalue of $A$ . Give an example for a $2 \times 2$ matrix, where the eigenvalues are complex.

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Problem 17.1.20

Let $V$ be an $n \times n$ normal matrix over $\mathbb {C}$ . Assume that all its eigenvalues have absolute value of $1$ , i.e. they are of the form $e^{i\phi }$ . Show that $V$ is unitary.

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Problem 17.1.21

Find square roots of the Pauli spin matrices

\sigma _{1} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad \sigma _{2} = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad \sigma _{3} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.

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Problem 17.1.22

Can one find a unitary matrix $U$ with $\operatorname {det}\left(U\right) = 1$ , i.e. $U$ is an element of $SU(2)$ , such that

\begin{pmatrix} 1 \\ 0 \end{pmatrix} = U \begin{pmatrix} 0 \\ 1 \end{pmatrix} ?

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Problem 17.1.23

Can one $2 \times 2$ find unitary matrices such that the rows and columns add up to one? Of course the $2 \times 2$ identity matrix is one of them.

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Problem 17.1.24

Let $U$ be an $n \times n$ unitary matrix. Then

\operatorname {det}\left(U-\lambda I_{n}\right) = \prod _{j=1}^{N} (e^{i\phi _{j}}-\lambda )

where $\phi_{j} \in [0,2\pi )$ . Let $n = 2$ . Find $\lambda$ for the matrix

U = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}.

Then calculate $\left|e^{i\phi_{1}}-e^{i\phi_{2}}\right|$ .

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Problem 17.1.25

Find a unitary $2 \times 2$ matrix such that $\sigma_{2}U-U\sigma_{3} = 0_{2}$ .

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Problem 17.1.26

The Pauli spin matrices $\sigma_{1}, \sigma_{2}, \sigma_{3}$ are hermitian and unitary. Together with the $2 \times 2$ identity matrix $\sigma_{0} = I_{2}$ they form an orthogonal basis in Hilbert space of the $2 \times 2$ matrices over $\mathbb {C}$ with the scalar product $\operatorname {tr}(AB^{*})$ . Let $X$ be an $n \times n$ hermitian matrix. Then $(X+iI_{n})^{-1}$ exists and

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

is unitary. This is the Cayley transform of $X$ . Find the Cayley transform of the Pauli spin matrix $\sigma_{2}$ .

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Problem 17.1.27

(i)(ii)

Let $A$ be an $n \times n$ hermitian matrix. Then $(A+iI_{n})^{-1}$ exists and

U = (A-iI_{n})(A+iI_{n})^{-1} \tag {1}

is a unitary matrix (Cayley transform of $A$ ).

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(i)

Show that $+1$ cannot be an eigenvalue of $U$ .

(ii)

Show that $A = i(U+I_{n})(I_{n}-U)^{-1}$ .

Problem 17.1.28

Find the Cayley transform of the hermitian matrix

H = \begin{pmatrix} h_{11} & h_{12} \\ \overline{h_{12}} & h_{22} \end{pmatrix}, \qquad h_{11}, h_{22} \in \mathbb {R}, \quad h_{12} \in \mathbb {C}.

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Problem 17.1.29

Let $\delta_{j}, \eta_{j} \in \mathbb {R}$ with $j = 1,2,3$ . Any $3 \times 3$ unitary symmetric matrix $U$ can be written in the product form

U = \begin{pmatrix} e^{i\delta _{1}} & 0 & 0 \\ 0 & e^{i\delta _{2}} & 0 \\ 0 & 0 & e^{i\delta _{3}} \end{pmatrix} \begin{pmatrix} \eta _{1} & \gamma _{12} & \gamma _{13} \\ \gamma _{12} & \eta _{2} & \gamma _{23} \\ \gamma _{13} & \gamma _{23} & \eta _{3} \end{pmatrix} \begin{pmatrix} e^{i\delta _{1}} & 0 & 0 \\ 0 & e^{i\delta _{2}} & 0 \\ 0 & 0 & e^{i\delta _{3}} \end{pmatrix}

where $\gamma_{jk} = N_{jk}\mathrm{exp}\left(i\beta_{jk}\right)$ with $N_{jk}, \beta_{jk} \in \mathbb {R}$ . It follows that

U_{jj} = \eta _{j}\mathrm{exp}\left(2i\delta _{j}\right), \qquad U_{jk} = N_{jk}\mathrm{exp}\left(i(\delta _{j}+\delta _{k}+\beta _{jk})\right).

The unitary condition $UU^{*} = I_{3}$ provides

\sum _{k \neq j}^{3} N_{jk}^{2}+\eta _{j} = 1, \qquad j = 1,2,3

and

N_{12}(\eta _{1}\mathrm{exp}\left(i\beta _{12}\right)+\eta _{2}\mathrm{exp}\left(-i\beta _{12}\right)) = N_{13}N_{23}\mathrm{exp}\left(i(\pi +\beta _{23}-\beta _{13})\right)

and cyclic ( $1 \to 2 \to 3 \to 1$ ). Write the unitary symmetric matrix

W = \begin{pmatrix} 0 & 0 & i \\ 0 & i & 0 \\ i & 0 & 0 \end{pmatrix}

in this form.

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Problem 17.1.30

(i)(ii)

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(i)

Consider the matrix

R = \begin{pmatrix} 1/2 & 1/2 & 1/\sqrt{2} \\ 1/2 & 1/2 & -1/\sqrt{2} \\ 1/\sqrt{2} & -1/\sqrt{2} & 0 \end{pmatrix}.

Show that $R^{-1} = R^{*} = R$ . Use these properties and $\operatorname {tr}(R)$ to find all the eigenvalues of $R$ . Find the eigenvectors.

(ii)

Let

A_{1} = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}, \qquad A_{2} = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 1 \\ 1 & 1 & 0 \end{pmatrix}.

Calculate $RA_{1}R^{-1}$ and $RA_{2}R^{-1}$ . Discuss.