Math Problems Vault

Problem 9.1.1

Consider the $2 \times 2$ nonnormal matrix

A = \begin{pmatrix} 3 & -4 \\ 1 & -1 \end{pmatrix}.

Calculate $A^n$ , where $n \in \mathbb {N}$ .

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Problem 9.1.2

Consider the $2 \times 2$ matrix $A = (a_{jk})$ over the complex numbers. Find $\mathrm{exp}\left(A\right)$ .

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Problem 9.1.3

(i)(ii)

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(i)

Consider the $2 \times 2$ matrix

A = \begin{pmatrix} a_{11} & a_{12} \\ 0 & a_{22} \end{pmatrix}.

Find $\mathrm{exp}\left(tA\right)$ .

(ii)

Let $\alpha , \beta \in \mathbb {R}$ and

M(\alpha ,\beta ) = \begin{pmatrix} \alpha & -\beta \\ \beta & \alpha \end{pmatrix}.

Calculate $\mathrm{exp}\left(M(\alpha ,\beta )\right)$ .

Problem 9.1.4

Consider the $2 \times 2$ unitary matrix

U = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}.

Can we find an $\alpha \in \mathbb {R}$ such that $U = \mathrm{exp}\left(\alpha A\right)$ , where

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

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Problem 9.1.5

(i)(ii)

Let $H$ be a hermitian matrix, i.e. $H = H^*$ . Now $U := e^{iH}$ is a unitary matrix. Let

H = \begin{pmatrix} a & b \\ \overline{b} & a \end{pmatrix}, \qquad a \in \mathbb {R}, \; b \in \mathbb {C}

with $b \neq 0$ .

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(i)

Calculate $e^{iH}$ using the normalized eigenvectors of $H$ to construct a unitary matrix $V$ such that $V^*HV$ is a diagonal matrix.

(ii)

Specify $a$ , $b$ such that we find the unitary matrix

U = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}.

Problem 9.1.6

Let $z \in \mathbb {C}$ . Let $A$ , $B$ be $n \times n$ matrices over $\mathbb {C}$ . We say that $B$ is invariant with respect to $A$ if

e^{zA}Be^{-zA} = B.

Obviously $e^{-zA}$ is the inverse of $e^{zA}$ . Show that, if this condition is satisfied, one has $[A,B] = 0_n$ . If $e^{zA}$ would be unitary we have $UBU^* = B$ .

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Problem 9.1.7

(i)(ii)(iii)

Let $z \in \mathbb {C}$ .

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(i)

Consider the $2 \times 2$ matrices

A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad B = \begin{pmatrix} b_{11} & b_{12} \\ b_{12} & b_{11} \end{pmatrix}.

Calculate $\mathrm{exp}\left(zA\right)$ , $\mathrm{exp}\left(-zA\right)$ and $\mathrm{exp}\left(zA\right)B\mathrm{exp}\left(-zA\right)$ . Calculate the commutator $[A,B]$ .

(ii)

Consider the $2 \times 2$ matrices

C = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad D = \begin{pmatrix} d_{11} & d_{12} \\ -d_{12} & d_{11} \end{pmatrix}.

Calculate $\mathrm{exp}\left(zC\right)$ , $\mathrm{exp}\left(-zC\right)$ and $\mathrm{exp}\left(zC\right)D\mathrm{exp}\left(-zC\right)$ . Calculate the commutator $[C,D]$ .

(iii)

Consider the $2 \times 2$ matrices

E = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \qquad F = \begin{pmatrix} f_{11} & f_{12} \\ 0 & f_{11} \end{pmatrix}.

Calculate $\mathrm{exp}\left(zE\right)$ , $\mathrm{exp}\left(-zE\right)$ and $\mathrm{exp}\left(zE\right)F\mathrm{exp}\left(-zE\right)$ . Calculate the commutator $[E,F]$ .

Problem 9.1.8

Let $\sigma_1$ , $\sigma_2$ , $\sigma_3$ be the Pauli spin matrices. Consider the $2 \times 2$ matrix

U(\alpha ,\beta ,\gamma ) = e^{-i\alpha \sigma _3/2}e^{-i\beta \sigma _2/2}e^{-i\gamma \sigma _3/2}

where $\alpha$ , $\beta$ , $\gamma$ are the three Euler angles with the range $0 \leq \alpha < 2\pi$ , $0 \leq \beta \leq \pi$ and $0 \leq \gamma < 2\pi$ . Show that

U(\alpha ,\beta ,\gamma ) = \begin{pmatrix} e^{-i\alpha /2}\cos (\beta /2)e^{-i\gamma /2} & -e^{-i\alpha /2}\sin (\beta /2)e^{i\gamma /2} \\ e^{-i\alpha /2}\sin (\beta /2)e^{-i\gamma /2} & e^{i\alpha /2}\cos (\beta /2)e^{i\gamma /2} \end{pmatrix}. \tag {1}

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Problem 9.1.9

(i)(ii)(iii)

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(i)

Let $\alpha \in \mathbb {R}$ . Consider the $3 \times 3$ symmetric matrix

A(\alpha ) = \begin{pmatrix} 0 & 0 & \alpha \\ 0 & 0 & 0 \\ \alpha & 0 & 0 \end{pmatrix}.

Find $\mathrm{exp}\left(A(\alpha )\right)$ .

(ii)

Consider the $3 \times 3$ matrix

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

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

(iii)

Let $a_{12}, a_{13}, a_{23} \in \mathbb {R}$ . Consider the skew-symmetric $3 \times 3$ matrix

A(a_{12},a_{13},a_{23}) = \begin{pmatrix} 0 & a_{12} & a_{13} \\ -a_{12} & 0 & a_{23} \\ -a_{13} & -a_{23} & 0 \end{pmatrix}.

Find $\mathrm{exp}\left(A\right)$ .

Problem 9.1.10

(i)(ii)

Let $A$ be an $n \times n$ matrix over $\mathbb {C}$ with $A^2 = rA$ , where $r \in \mathbb {C}$ and $r \neq 0$ .

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(i)

Calculate $e^{zA}$ , where $z \in \mathbb {C}$ .

(ii)

Let $U(z) = e^{zA}$ . Let $z' \in \mathbb {C}$ . Calculate $U(z)U(z')$ .

Problem 9.1.11

Can we find a $2 \times 2$ matrix $B$ over the real numbers $\mathbb {R}$ such that

\sin (B) = \begin{pmatrix} 1 & 4 \\ 0 & 1 \end{pmatrix}? \tag {1}

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Problem 9.1.12

(i)(ii)

Let $A$ be an $n \times n$ matrix over $\mathbb {C}$ . Assume that $A^2 = cI_n$ , where $c \in \mathbb {R}$ .

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(i)

Calculate $\mathrm{exp}\left(A\right)$ .

(ii)

Apply the result to the $2 \times 2$ matrix ( $z \neq 0$ )

B = \begin{pmatrix} 0 & z \\ -\overline{z} & 0 \end{pmatrix}.

Thus $B$ is skew-hermitian, i.e. $\overline{B}^T = -B$ .

Problem 9.1.13

(i)(ii)

For any $n \times n$ matrix $A$ we have the identity

\operatorname {det}\left(\mathrm{exp}\left(A\right)\right) \equiv \mathrm{exp}\left(\operatorname {tr}(A)\right).

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(i)

Any $n \times n$ unitary matrix $U$ can be written as $U = \mathrm{exp}\left(iK\right)$ , where $K$ is a hermitian matrix. Assume that $\operatorname {det}\left(U\right) = -1$ . What can be said about the trace of $K$ ?

(ii)

Let

A = \begin{pmatrix} 2 & 3 \\ 7 & -2 \end{pmatrix}.

Calculate $\operatorname {det}\left(\mathrm{exp}\left(A\right)\right)$ .

Problem 9.1.14

Let $A$ be an $n \times n$ matrix. Then $\mathrm{exp}\left(A\right)$ can also be calculated as

e^A = \lim _{m \to \infty } \left(I_n + \frac{A}{m}\right)^m.

Use this definition to show that $\operatorname {det}\left(\mathrm{exp}\left(A\right)\right) \equiv \mathrm{exp}\left(\operatorname {tr}(A)\right)$ .

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Problem 9.1.15

Let $A$ , $B$ be $n \times n$ matrices. Then we have the identity

\operatorname {det}\left(e^Ae^Be^{-A}e^{-B}\right) \equiv \mathrm{exp}\left(\operatorname {tr}([A,B])\right).

Show that $\operatorname {det}\left(e^Ae^Be^{-A}e^{-B}\right) = 1$ .

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Problem 9.1.16

The MacLaurin series for $\arctan (z)$ is defined as

\arctan (z) = z - \frac{z^3}{3} + \frac{z^5}{5} - \frac{z^7}{7} + \cdots = \sum _{j=0}^{\infty } \frac{(-1)^j z^{2j+1}}{2j+1}

which converges for all complex values of $z$ having absolute value less than 1, i.e. $\left|z\right| < 1$ . Let $A$ be an $n \times n$ matrix. Thus the series expansion

\arctan (A) = A - \frac{A^3}{3} + \frac{A^5}{5} - \frac{A^7}{7} + \cdots = \sum _{j=0}^{\infty } \frac{(-1)^j A^{2j+1}}{2j+1}

is well-defined for $A$ if all eigenvalues $\lambda$ of $A$ satisfy $\left|\lambda \right| < 1$ . Let

A = \frac{1}{2} \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}.

Does $\arctan (A)$ exist?

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Problem 9.1.17

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

[A,[A,B]] = [B,[A,B]] = 0_n. \tag {1}

Show that

e^{A+B} = e^A e^B e^{-\frac{1}{2}[A,B]} \tag {2a}

e^{A+B} = e^B e^A e^{+\frac{1}{2}[A,B]}. \tag {2b}

Use the technique of parameter differentiation, i.e. consider the matrix-valued function

f(\epsilon ) := e^{\epsilon A}e^{\epsilon B}

where $\epsilon$ is a real parameter. Then take the derivative of $f$ with respect to $\epsilon$ .

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Problem 9.1.18

(i)(ii)

Let $A$ , $B$ , $C_2$ , , $C_m$ , be $n \times n$ matrices over $\mathbb {C}$ . The Zassenhaus formula is given by

\mathrm{exp}\left(A+B\right) = \mathrm{exp}\left(A\right)\mathrm{exp}\left(B\right)\mathrm{exp}\left(C_2\right) \cdots \mathrm{exp}\left(C_m\right) \cdots .

The left-hand side is called the disentangled form and the right-hand side is called the undisentangled form. Find $C_2, C_3, \ldots$ , using the comparison method. In the comparison method the disentangled and undisentangled forms are expanded in terms of an ordering scalar $\alpha$ and matrix coefficients of equal powers of $\alpha$ are compared. From

\exp (\alpha (A+B)) = \exp (\alpha A)\exp (\alpha B)\exp (\alpha ^2 C_2)\exp (\alpha ^3 C_3) \cdots

we obtain

\sum _{k=0}^{\infty } \frac{\alpha ^k}{k!}(A+B)^k = \sum _{r_0,r_1,r_2,r_3,\ldots =0}^{\infty } \frac{\alpha ^{r_0+r_1+2r_2+3r_3+\ldots }}{r_0!r_1!r_2!r_3!\cdots } A^{r_0}B^{r_1}C_2^{r_2}C_3^{r_3} \cdots .

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(i)

Find $C_2$ and $C_3$ .

(ii)

Assume that $[A,[A,B]] = 0_n$ and $[B,[A,B]] = 0_n$ . What conclusion can we draw for the Zassenhaus formula?

Problem 9.1.19

Let $A$ , $B$ be $n \times n$ matrices and $t \in \mathbb {R}$ . Show that

e^{t(A+B)} - e^{tA}e^{tB} = \frac{t^2}{2}(BA-AB) + \text{higher order terms in } t. \tag {1}

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Problem 9.1.20

Let $A$ be an $n \times n$ matrix with $A^3 = -A$ and $\alpha \in \mathbb {R}$ . Calculate $\exp (\alpha A)$ .

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Problem 9.1.21

(i)(ii)

Let $X$ be an $n \times n$ matrix over $\mathbb {C}$ . Assume that $X^2 = I_n$ . Let $Y$ be an arbitrary $n \times n$ matrix over $\mathbb {C}$ . Let $z \in \mathbb {C}$ .

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(i)

Calculate $\exp (zX)Y\exp (-zX)$ using the Baker-Campbell-Hausdorff formula

e^{zX}Ye^{-zX} = Y + z[X,Y] + \frac{z^2}{2!}[X,[X,Y]] + \frac{z^3}{3!}[X,[X,[X,Y]]] + \cdots .

(ii)

Calculate $\exp (zX)Y\exp (-zX)$ by first calculating $\exp (zX)$ and $\exp (-zX)$ and then doing the matrix multiplication. Compare the two methods.

Problem 9.1.22

Let $K$ be a hermitian matrix. Then $U := \mathrm{exp}\left(iK\right)$ is a unitary matrix. A method to find the hermitian matrix $K$ from the unitary matrix $U$ is to consider the principal logarithm of a matrix $A \in \mathbb {C}^{n \times n}$ with no eigenvalues on $\mathbb {R}^-$ (the closed negative real axis). This logarithm is denoted by $\log (A)$ and is the unique matrix $B$ such that $\mathrm{exp}\left(B\right) = A$ and the eigenvalues of $B$ have imaginary parts lying strictly between $-\pi$ and $\pi$ . For $A \in \mathbb {C}^{n \times n}$ with no eigenvalues on $\mathbb {R}^-$ we have the following integral representation

\log (s(A-I_n)+I_n) = \int _0^s (A-I_n)(t(A-I_n)+I_n)^{-1}dt.

Thus with $s=1$ we obtain

\log (A) = \int _0^1 (A-I_n)(t(A-I_n)+I_n)^{-1}dt.

Find $\log (U)$ of the unitary matrix

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

First test whether the method can be applied.

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Problem 9.1.23

(i)(ii)(iii)(iv)

Consider the matrix ( $\zeta \in \mathbb {R}$ )

S(\zeta ) = \begin{pmatrix} \cosh (\zeta ) & 0 & 0 & \sinh (\zeta ) \\ 0 & \cosh (\zeta ) & \sinh (\zeta ) & 0 \\ 0 & \sinh (\zeta ) & \cosh (\zeta ) & 0 \\ \sinh (\zeta ) & 0 & 0 & \cosh (\zeta ) \end{pmatrix}.

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(i)

Show that the matrix is invertible, i.e. find the determinant.

(ii)

Calculate the inverse of $S(\zeta )$ .

(iii)

Calculate

A := \frac{d}{d\zeta }S(\zeta )\bigg|_{\zeta =0}

and then calculate $\exp (\zeta A)$ .

(iv)

Do the matrices $S(\zeta )$ form a group under matrix multiplication?

Problem 9.1.24

Let $M$ be an $n \times n$ matrix with $m_{jk} = 1$ for all $j,k = 1,\ldots ,n$ . Let $z \in \mathbb {C}$ . Find $\mathrm{exp}\left(zM\right)$ . Then consider the special case $zn = i\pi$ .

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Problem 9.1.25

Let $X$ , $Y$ be $n \times n$ matrices. Show that

[e^X,Y] = \sum _{k=1}^{\infty } \frac{[X^k,Y]}{k!}.

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Problem 9.1.26

(i)(ii)

Let $A$ , $B$ be $n \times n$ matrices over $\mathbb {C}$ and $\alpha \in \mathbb {C}$ . The Baker-Campbell-Hausdorff formula states that

e^{\alpha A}Be^{-\alpha A} = B + \alpha [A,B] + \frac{\alpha ^2}{2!}[A,[A,B]] + \cdots = \sum _{j=0}^{\infty } \frac{\alpha ^j}{j!}\{ A^j,B\} = \widetilde{B}(\alpha )

where $[A,B] := AB-BA$ and $\{ A^j,B\} := [A,\{ A^{j-1},B\} ]$ is the repeated commutator.

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(i)

Extend the formula to $e^{\alpha A}B^ke^{-\alpha A}$ , where $k \geq 1$ .

(ii)

Extend the formula to $e^{\alpha A}e^Be^{-\alpha A}$ .

Problem 9.1.27

Consider the $n \times n$ matrix ( $n \geq 2$ )

A = \begin{pmatrix} 0 & 1 & 0 & \ldots & 0 \\ 0 & 0 & 1 & \ldots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \ldots & 1 \\ 0 & 0 & 0 & \ldots & 0 \end{pmatrix}.

Let $f: \mathbb {R} \to \mathbb {R}$ be an analytic function. Calculate

f(0)I_n + \frac{f'(0)}{1!}A + \frac{f''(0)}{2!}A^2 + \cdots + \frac{f^{n-1}(0)}{(n-1)!}A^{n-1}

where $'$ denotes differentiation. Discuss.

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Problem 9.1.28

Let $A$ , $B$ be $n \times n$ matrices with $A^2 = I_n$ and $B^2 = I_n$ . Assume that $[A,B] = 0_n$ . Let $a, b \in \mathbb {C}$ . Calculate $e^{aA+bB}$ .

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Problem 9.1.29

Let $A$ , $B$ be $n \times n$ matrices with $A^2 = I_n$ and $B^2 = I_n$ . Assume that $[A,B]_+ = 0_n$ . Let $a, b \in \mathbb {C}$ . Calculate $e^{aA+bB}$ .

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Problem 9.1.30

(i)(ii)

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(i)

Let $A$ , $B$ be $n \times n$ matrices over $\mathbb {C}$ . Assume that $A^2 = I_n$ and $B^2 = I_n$ . Find $\exp (\alpha A \otimes \beta B)$ , where $\alpha , \beta \in \mathbb {C}$ .

(ii)

Let $A$ , $B$ , $C$ be $n \times n$ matrices over $\mathbb {C}$ such that $A^2 = I_n$ , $B^2 = I_n$ and $C^2 = I_n$ . Furthermore assume that

[A,B]_+ = 0_n, \qquad [B,C]_+ = 0_n, \qquad [C,A]_+ = 0_n.

Let $\alpha , \beta , \gamma \in \mathbb {C}$ . Calculate $e^{\alpha A+\beta B+\gamma C}$ using

e^{\alpha A+\beta B+\gamma C} = \sum _{j=0}^{\infty } \frac{(\alpha A+\beta B+\gamma C)^j}{j!}.

Problem 9.1.31

(i)(ii)

Let $A$ be an $n \times n$ matrix over $\mathbb {C}$ . Assume that all eigenvalues $\lambda_1, \ldots , \lambda_n$ are pairwise distinct.

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(i)

Then $e^{tA}$ can be calculated as follows (Lagrange interpolation)

e^{tA} = \sum _{j=1}^{n} e^{\lambda _j t} \prod _{\substack {k=1 \\ k \neq j}}^{n} \frac{(A-\lambda _k I_n)}{(\lambda _j-\lambda _k)}.

Consider the Pauli spin matrix $\sigma_1$ . Calculate $e^{tA}$ using this method.

(ii)

$e^{tA}$ can also be calculated as follows (Newton interpolation)

e^{tA} = e^{\lambda _1 t}I_n + \sum _{j=2}^{n} [\lambda _1,\ldots ,\lambda _j] \prod _{k=1}^{j-1} (A-\lambda _k I_n).

The divided differences $[\lambda_1,\ldots ,\lambda_j]$ depend on $t$ and are defined recursively by

[\lambda _1,\lambda _2] := \frac{e^{\lambda _1 t}-e^{\lambda _2 t}}{\lambda _1-\lambda _2}

[\lambda _1,\ldots ,\lambda _{k+1}] := \frac{[\lambda _1,\ldots ,\lambda _k]-[\lambda _2,\ldots ,\lambda _{k+1}]}{\lambda _1-\lambda _{k+1}}, \quad k \geq 2.

Calculate $e^{t\sigma_1}$ using this method.

Problem 9.1.32

Let $A$ be an $n \times n$ matrix. The characteristic polynomial

\det (\lambda I_n - A) = \lambda ^n + a_1\lambda ^{n-1} + \cdots + a_n = p(\lambda )

is closely related the resolvent $(\lambda I_n - A)^{-1}$ trough the formula

(\lambda I_n - A)^{-1} = \frac{N_1\lambda ^{n-1}+N_2\lambda ^{n-2}+\cdots +N_n}{\lambda ^n+a_1\lambda ^{n-1}+\cdots +a_n} = \frac{N(\lambda )}{p(\lambda )}

where the adjugate matrix $N(\lambda )$ is a polynomial in $\lambda$ of degree $n-1$ with constant $n \times n$ coefficient matrices $N_1,\ldots ,N_n$ . The Laplace transform of the matrix exponential is the resolvent

\mathcal{L}(e^{tA}) = (\lambda I_n - A)^{-1}.

The $N_k$ matrices and $a_k$ coefficients may be computed recursively as follows

N_1 = I_n, \qquad a_1 = -\frac{1}{1}\operatorname {tr}(AN_1)

N_2 = AN_1 + a_1I_n, \qquad a_2 = -\frac{1}{2}\operatorname {tr}(AN_2)

\vdots

N_n = AN_{n-1} + a_{n-1}I_n, \qquad a_n = -\frac{1}{n}\operatorname {tr}(AN_n)

0_n = AN_n + a_nI_n.

Consider the $2 \times 2$ unitary matrix

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

Find the $N_k$ matrices and the coefficients $a_k$ and thus calculate the resolvent.

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Problem 9.1.33

(i)(ii)

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(i)

Let $A$ , $B$ be $n \times n$ matrices over $\mathbb {C}$ . Calculate $e^ABe^A$ . Set $f(\epsilon ) = e^{\epsilon A}Be^{\epsilon A}$ , where $\epsilon$ is a real parameter. Then differentiate with respect to $\epsilon$ . For $\epsilon = 1$ we have $e^ABe^A$ .

(ii)

Let $\epsilon \in \mathbb {R}$ . Calculate the matrix-valued function

f(\epsilon ) = e^{-\epsilon \sigma _2}\sigma _3 e^{\epsilon \sigma _2}.

Differentiate the matrix-valued function $f$ with respect to $\epsilon$ and solve the initial value problem of the resulting ordinary differential equation.

Problem 9.1.34

Let $A$ be an $n \times n$ matrix over $\mathbb {C}$ . Let $T$ be a nilpotent matrix over $\mathbb {C}$ satisfying $T^*A + AT = 0_n$ . Show that $(e^T)^*Ae^T = A$ .

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Problem 9.1.35

(i)(ii)

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(i)

Let $\Pi$ be an $n \times n$ projection matrix. Let $\epsilon \in \mathbb {R}$ . Calculate $\exp (\epsilon \Pi )$ .

(ii)

Let $\Pi_1, \Pi_2, \ldots , \Pi_n$ be $n \times n$ projection matrices. Assume that $\Pi_j\Pi_k = 0_n$ ( $j \neq k$ ) for all $j,k = 1,\ldots ,n$ . Let $\epsilon_j \in \mathbb {R}$ with $j = 1,\ldots ,n$ . Calculate

\exp (\epsilon _1\Pi _1+\epsilon _2\Pi _2+\cdots +\epsilon _n\Pi _n).

Assume additionally that $\Pi_1+\Pi_2+\cdots +\Pi_n = I_n$ . Simplify the result from (i) using this condition.

Problem 9.1.36

Let $A$ , $B$ be $n \times n$ matrices over $\mathbb {C}$ . Assume that $A$ and $B$ commute with the commutator $[A,B]$ . Then

\exp (A+B) = \exp (A)\exp (B)\exp \left(-\frac{1}{2}[A,B]\right).

Can this formula be applied to the matrices

C = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \qquad D = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}.

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Problem 9.1.37

Let $\epsilon \in \mathbb {R}$ . Let $A$ , $B$ be $n \times n$ matrices over $\mathbb {C}$ . Expand $e^{\epsilon A}e^{\epsilon B}e^{-\epsilon A}e^{-\epsilon B}$ up to second order in $\epsilon$ .

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Problem 9.1.38

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

A(\alpha ) = \begin{pmatrix} \cos (\alpha ) & \sin (\alpha ) \\ \sin (\alpha ) & -\cos (\alpha ) \end{pmatrix}.

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(i)

Show that the matrix is orthogonal.

(ii)

Find the determinant of $A(\alpha )$ . Is the matrix an element of $SO(2,\mathbb {R})$ ?

(iii)

Do these matrices form a group under matrix multiplication?

(iv)

Calculate

X = \frac{d}{d\alpha }A(\alpha )\bigg|_{\alpha =0}.

Calculate $\exp (\alpha X)$ and compare this matrix with $A(\alpha )$ . Discuss.

(v)

Let $\beta \in \mathbb {R}$ and

B(\beta ) = \begin{pmatrix} \cos (\beta ) & \sin (\beta ) \\ \sin (\beta ) & -\cos (\beta ) \end{pmatrix}.

Is the matrix $A(\alpha ) \otimes B(\beta )$ orthogonal? Find the determinant of $A(\alpha ) \otimes A(\alpha )$ . Is this matrix an element of $SO(4,\mathbb {R})$ ?

Problem 9.1.39

Let $A$ be a normal matrix with eigenvalues $\lambda_1, \ldots , \lambda_n$ and corresponding normalized pairwise orthogonal eigenvectors $\overrightarrow {u}_1, \ldots , \overrightarrow {u}_n$ . Let $\overrightarrow {w}, \overrightarrow {v} \in \mathbb {C}^{n}$ (column vectors). Find $\overrightarrow {w}^* \mathrm{exp}\left(A\right)\overrightarrow {v}$ by expanding $\overrightarrow {w}$ and $\overrightarrow {v}$ with respect to the basis $\overrightarrow {u}_j$ ( $j = 1,\ldots ,n$ ).

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Problem 9.1.40

Let $B$ be an $n \times n$ matrix with $B^2 = I_n$ . Show that

\exp \left(-\frac{1}{2}i\pi (B-I_n)\right) \equiv B.

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Problem 9.1.41

Consider the $2 \times 2$ nonnormal matrices

A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \qquad B = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} = A^*.

Are the matrices

\mathrm{exp}\left(A\right), \qquad \mathrm{exp}\left(B\right)

normal? Are the matrices $\sin (A)$ , $\sin (B)$ , $\cos (A)$ , $\cos (B)$ normal?

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Problem 9.1.42

Let $z \in \mathbb {C}$ . Let $A$ be an $n \times n$ matrix over $\mathbb {C}$ . Assume that $A^3 = zA$ . Find $\mathrm{exp}\left(A\right)$ .

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Problem 9.1.43

Let $\overrightarrow {v}_1$ , $\overrightarrow {v}_2$ be an orthonormal set in $\mathbb {C}^{2}$ . Consider the $2 \times 2$ matrix

A = -i\overrightarrow {v}_1\overrightarrow {v}_1^* + i\overrightarrow {v}_2\overrightarrow {v}_2^*.

Thus $I_2 = \overrightarrow {v}_1\overrightarrow {v}_1^* + \overrightarrow {v}_2\overrightarrow {v}_2^*$ . Find $K$ such that $\mathrm{exp}\left(K\right) = A$ .

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Problem 9.1.44

(i)(ii)

Any $2 \times 2$ matrix $A$ can be written as a linear combination of the Pauli spin matrices and the $2 \times 2$ identity matrix $I_2$

A = aI_2 + b\sigma _1 + c\sigma _2 + d\sigma _3

where $a,b,c,d \in \mathbb {C}$ .

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(i)

Find $A^2$ and $A^3$ .

(ii)

Use the result from (i) to find all matrices $A$ such that $A^3 = \sigma_1$ , i.e. calculate the third root $\sqrt[3]{\sigma_1}$ .

Problem 9.1.45

Let $A$ , $B$ , $H$ , $K$ be $n \times n$ matrices over $\mathbb {C}$ . Assume that $A^2 = B^2 = 0_n$ . Let $z \in \mathbb {C}$ . Calculate

e^{z(A \otimes B)}(H \otimes K)e^{-z(A \otimes B)}

using the Baker-Campbell-Hausdorff formula.

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Problem 9.1.46

(i)(ii)

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(i)

Let $A$ , $B$ be $n \times n$ matrices over $\mathbb {C}$ , $z \in \mathbb {C}$ and $\overrightarrow {u}, \overrightarrow {v} \in \mathbb {C}^{n}$ . Calculate

\exp (z(A \otimes B))(\overrightarrow {u} \otimes \overrightarrow {v}).

(ii)

Simplify the result from (i) if $A^2 = I_n$ and $B^2 = I_n$ .

Problem 9.1.47

Let $A$ be an $n \times n$ matrix. Calculate

\exp \begin{pmatrix} 0_n & A \\ A & 0_n \end{pmatrix}

using the identity

\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \otimes A \equiv \begin{pmatrix} 0_n & A \\ A & 0_n \end{pmatrix}.

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Problem 9.1.48

(i)(ii)

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(i)

Let $A_1, A_2, \ldots , A_m$ be $n \times n$ matrices with $A_j^2 = I_n$ for $j = 1,\ldots ,m$ . Let $z \in \mathbb {C}$ . Calculate

\exp (z(A_1 \otimes A_2 \otimes \cdots \otimes A_m)).

(ii)

Use the result from (i) to calculate $\exp (-i\theta (\sigma_1 \otimes \sigma_2 \otimes \sigma_3))$ , where $\theta \in \mathbb {R}$ .

Problem 9.1.49

(i)(ii)(iii)

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(i)

Let $A$ be an $n \times n$ matrix over $\mathbb {C}$ and $\Pi$ be an $m \times m$ projection matrix. Let $z \in \mathbb {C}$ . Calculate

\exp (z(A \otimes \Pi )).

(ii)

Let $A_1$ , $A_2$ be $n \times n$ matrices over $\mathbb {C}$ . Let $\Pi_1$ , $\Pi_2$ be $m \times m$ projection matrices with $\Pi_1\Pi_2 = 0_n$ . Calculate

\exp (z(A_1 \otimes \Pi _1 + A_2 \otimes \Pi _2)).

(iii)

Use the result from (ii) to find the unitary matrix

U(t) = \exp (-i\hat{H}t/\hbar )

where $\hat{H} = \hbar \omega (A_1 \otimes \Pi_1 + A_2 \otimes \Pi_2)$ and we assume that $A_1$ and $A_2$ are hermitian matrices.

Problem 9.1.50

(i)(ii)(iii)

Let $A$ , $B$ be $n \times n$ matrices over $\mathbb {C}$ . Let $\overrightarrow {u}$ be an eigenvector of $A$ and $\overrightarrow {v}$ be an eigenvector of $B$ , respectively, i.e. $A\overrightarrow {u} = \lambda \overrightarrow {u}$ , $B\overrightarrow {v} = \mu \overrightarrow {v}$ .

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(i)

Find $e^{A \otimes B}(\overrightarrow {u} \otimes \overrightarrow {v})$ .

(ii)

Find $e^{A \otimes I_n+I_n \otimes B}(\overrightarrow {u} \otimes \overrightarrow {v})$ .

(iii)

Find $e^{A \otimes I_n+I_n \otimes B+A \otimes B}(\overrightarrow {u} \otimes \overrightarrow {v})$ .

Problem 9.1.51

For every positive definite matrix $A$ , there is a unique positive definite matrix $Q$ such that $Q^2 = A$ . The matrix $Q$ is called the square root of $A$ . Can we find the square root of the matrix

B = \frac{1}{2} \begin{pmatrix} 5 & 3 \\ 3 & 5 \end{pmatrix}?

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Problem 9.1.52

Consider the $2 \times 2$ matrix

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

The matrix is symmetric over $\mathbb {R}$ and the eigenvalues are positive. Thus the matrix is positive definite. Show that

X = \frac{1}{\sqrt{5}} \begin{pmatrix} 2 & 1 \\ 1 & 3 \end{pmatrix}, \qquad Y = \begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix}

are square roots of $A$ . Discuss.

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Problem 9.1.53

Find the square root of the Pauli matrix $\sigma_1$ applying the spectral theorem.

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Problem 9.1.54

Let $A$ be an $n \times n$ matrix over $\mathbb {C}$ . The $n \times n$ matrix $B$ over $\mathbb {C}$ is a square root of $A$ iff $B^2 = A$ . The number of square roots of a given matrix $A$ may be zero, finite or infinite. Does the matrix

A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}

admit a square root?

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Problem 9.1.55

Find a square root of the nonnormal $3 \times 3$ matrix

A = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}

i.e. find the matrices $B$ such that $B^2 = A$ .

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Problem 9.1.56

Consider the two-dimensional rotation matrix

R(\alpha ) = \begin{pmatrix} \cos (\alpha ) & -\sin (\alpha ) \\ \sin (\alpha ) & \cos (\alpha ) \end{pmatrix}

with $0 \leq \alpha \leq \pi$ . Find a square root $R^{1/2}(\alpha )$ of $R(\alpha )$ , i.e. a matrix $S$ such that $S^2 = R(\alpha )$ .

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Problem 9.1.57

Let $V_1$ be a hermitian $n \times n$ matrix. Let $V_2$ be a positive semidefinite $n \times n$ matrix. Let $k$ be a positive integer. Show that

\operatorname {tr}((V_2V_1)^k)

can be written as $\operatorname {tr}(V^k)$ , where $V := V_2^{1/2}V_1V_2^{1/2}$ .

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Problem 9.1.58

Let $A$ , $B$ be $n \times n$ hermitian matrices. Then (Golden-Thompson inequality)

\operatorname {tr}(e^{A+B}) \leq \operatorname {tr}(e^Ae^B).

Let $A = \sigma_3$ and $B = \sigma_1$ . Calculate the left and right-hand side of the inequality.

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Problem 9.1.59

Let $A$ be an $n \times n$ matrix over $\mathbb {R}$ . The $(p,q)$ Pade approximation to $\mathrm{exp}\left(A\right)$ is defined by

R_{pq}(A) := (D_{pq}(A))^{-1}N_{pq}(A)

where

N_{pq}(A) = \sum _{j=0}^{p} \frac{(p+q-j)!p!}{(p+q)!j!(p-j)!}A^j, \qquad D_{pq}(A) = \sum _{j=0}^{q} \frac{(p+q-j)!q!}{(p+q)!j!(q-j)!}(-A)^j.

Nonsingularity of $D_{pq}(A)$ is assured if $p$ and $q$ are large enough or if the eigenvalues of $A$ are negative. Find the Pade approximation for the matrix

A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}

and $p = q = 2$ . Compare with the exact solution.

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Problem 9.1.60

Let $A$ be an $n \times n$ matrix. We define the $j-k$ approximant of $\mathrm{exp}\left(A\right)$ by

f_{j,k}(A) := \left(\sum _{\ell =0}^{k} \frac{1}{\ell !}\left(\frac{A}{j}\right)^{\ell }\right)^j. \tag {1}

We have the inequality

\left\| e^A-f_{j,k}(A)\right\| \leq \frac{1}{j^k(k+1)!}\left\| A\right\| ^{k+1}e^{\left\| A\right\| } \tag {2}

and $f_{j,k}(A)$ converges to $e^A$ , i.e.

\lim _{j \to \infty } f_{j,k}(A) = \lim _{k \to \infty } f_{j,k}(A) = e^A.

Let

A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}.

Find $f_{2,2}(A)$ and $e^A$ . Calculate the right-hand side of the inequality (2).

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Problem 9.1.61

(i)(ii)

The Denman-Beavers iteration for the square root of an $n \times n$ matrix $A$ with no eigenvalues on $\mathbb {R}^-$ is

Y_{k+1} = \frac{1}{2}(Y_k+Z_k^{-1}), \qquad Z_{k+1} = \frac{1}{2}(Z_k+Y_k^{-1})

with $k = 0,1,2,\ldots$ and $Z_0 = I_n$ and $Y_0 = A$ . The iteration has the properties that

\lim _{k \to \infty } Y_k = A^{1/2}, \qquad \lim _{k \to \infty } Z_k = A^{-1/2}

and, for all $k$ ,

Y_k = AZ_k, \qquad Y_kZ_k = Z_kY_k, \qquad Y_{k+1} = \frac{1}{2}(Y_k+AY_k^{-1}).

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(i)

Show that the Denman-Beavers iteration be applied to the matrix

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

(ii)

Find $Y_1$ and $Z_1$ .