Math Problems Vault

Problem 9.2.1

Let $z_1, z_2, z_3 \in \mathbb {C}$ . Consider the $2 \times 2$ matrix

A(z_1,z_2,z_3) = \begin{pmatrix} z_1 & z_2 \\ z_3 & 0 \end{pmatrix}.

Calculate $\mathrm{exp}\left(A(z_1,z_2,z_3)\right)$ .

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Problem 9.2.2

Let $\alpha \in \mathbb {R}$ . Calculate the vector in $\mathbb {R}^{2}$

\exp \left(\alpha \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}\right) \begin{pmatrix} 1 \\ 0 \end{pmatrix}

in two different ways. Compare and discuss.

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Problem 9.2.3

(i)(ii)

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

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(i)

Show that

[e^A,e^B] = e^Ae^B - e^Be^A = \sum _{j,k=1}^{\infty } \frac{1}{j!k!}[A^j,B^k].

(ii)

Assume that for all eigenvalues $\lambda$ we have $\Re (\lambda ) < 0$ . Let $B$ be an arbitrary $n \times n$ matrix over $\mathbb {C}$ . Let

R(t) := \int _0^{\infty } e^{tA^*}Be^{tA}dt.

Show that the matrix $R$ satisfies $R(t)A + A^*R(t) = -B$ .

Problem 9.2.4

Let $A$ , $B$ be $n \times n$ matrices over $\mathbb {C}$ . Is $\operatorname {tr}(e^A \otimes e^B) = \operatorname {tr}(e^{A \otimes B})$ ? Prove or disprove.

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Problem 9.2.5

(i)(ii)

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(i)

Let $A_1, A_2, \ldots , A_p$ be $n \times n$ matrices over $\mathbb {C}$ . The generalized Trotter formula is given by

\exp \left(\sum _{j=1}^{n} A_j\right) = \lim _{n \to \infty } f_n(\left\{ A_j\right\} ) \tag {1}

where the $n$ -th approximant $f_n(\left\{ A_j\right\} )$ is defined by

f_n(\left\{ A_j\right\} ) := \left(\exp \left(\frac{1}{n}A_1\right)\exp \left(\frac{1}{n}A_2\right) \cdots \exp \left(\frac{1}{n}A_p\right)\right)^n.

Let $p = 2$ and

A_1 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \qquad A_2 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}.

Calculate the left and right-hand side of (1).

(ii)

Let $A$ , $B$ be $n \times n$ matrices over $\mathbb {C}$ with $A^2 = B^2 = I_n$ and $[A,B]_+ = 0_n$ . Let $z \in \mathbb {C}$ . The Lie-Trotter formula is given by

\exp (z(A+B)) = \lim _{p \to \infty } \left(e^{zA/p}e^{zB/p}\right)^p.

Calculate $e^{z(A+B)}$ using the right-hand side. Note that

e^{zA/p} = I_n\cosh (z/p) + A\sinh (z/p), \qquad e^{zB/p} = I_n\cosh (z/p) + B\sinh (z/p).

Problem 9.2.6

(i)(ii)

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(i)

Let $A$ , $B$ be $n \times n$ matrices with $AB = BA$ . Show that

\cos (A+B) \equiv \cos (A)\cos (B) - \sin (A)\sin (B).

(ii)

Let $x, y \in \mathbb {R}$ . We know that $\sin (x+y) \equiv \sin (x)\cos (y) + \cos (x)\sin (y)$ . Let

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

Is $\sin (C+D) = \sin (C)\cos (D) + \cos (C)\sin (D)$ ? Prove or disprove.

Problem 9.2.7

(i)(ii)

Let $A$ , $B$ be $n \times n$ matrices over $\mathbb {C}$ . The Laplace transform $\mathcal{L}$ of the matrix exponential is the resolvent

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

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(i)

Calculate $\mathcal{L}(e^{tA} \otimes e^{tB})$ .

(ii)

Let $\mathcal{L}^{-1}$ be the inverse Laplace transform. Find

\mathcal{L}^{-1}((\lambda I_n-A)^{-1} \otimes (\lambda I_n-B)^{-1}).

Problem 9.2.8

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.

Show that

\operatorname {tr}(\mathcal{L}(e^{tA})) = \frac{1}{p(\lambda )}\frac{dp(\lambda )}{d\lambda }.

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Problem 9.2.9

Let $A$ be an arbitrary $n \times n$ matrix. Can we conclude that $\mathrm{exp}\left(A^*\right) = (\mathrm{exp}\left(A\right))^*$ ?

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Problem 9.2.10

Let $A$ be an $n \times n$ matrix over $\mathbb {C}$ . Assume that $A$ is hermitian, i.e. $A^* = A$ . Thus $A$ has only real eigenvalues. Assume that $A^5+A^3+A = 3I_n$ . Show that $A = I_n$ .

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Problem 9.2.11

Let $A$ , $B$ be $n \times n$ hermitian matrices. There exists $n \times n$ unitary matrices $U$ and $V$ (depending on $A$ and $B$ ) such that

\mathrm{exp}\left(iA\right)\mathrm{exp}\left(iB\right) = \mathrm{exp}\left(iUAU^{-1}+iVBV^{-1}\right).

Consider $n = 2$ and

A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad B = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} \; \Rightarrow \; [A,B] = \sqrt{2} \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}.

Find $U$ and $V$ . Note that $A$ and $B$ are also unitary.

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Problem 9.2.12

Let $A$ , $B$ be $n \times n$ matrices over $\mathbb {C}$ such that $[A,B] = A$ . What can be said about the commutator $[e^A,e^B]$ ?

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Problem 9.2.13

(i)(ii)

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(i)

Let $A$ be an $n \times n$ nilpotent matrix. Is the matrix $\cos (A)$ invertible?

(ii)

Let $N$ be a nilpotent $n \times n$ matrix with $N^j = 0_n$ with $j \geq 1$ . Is

\ln (I_n-N) = -\left(N+\frac{1}{2}N+\cdots +\frac{1}{j-1}N^{j-1}\right)

and $\exp (\ln (I_n-N)) = I_n-N$ .

Problem 9.2.14

Let $H = H_1+H_2+H_3$ and $H_1$ , $H_2$ , $H_3$ be $n \times n$ hermitian matrices. Show that

e^{-\beta H} = e^{-\beta H_1/2}e^{-\beta H_2/2}e^{-\beta H_3}e^{-\beta H_2/2}e^{-\beta H_1/2} + \left(-\frac{1}{2}\beta \right)^3 S

where

S = \frac{1}{6}([[H_2+H_3,H_1],H_1+2H_2+2H_3] + [[H_3,H_2],H_2+H_3]).

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Problem 9.2.15

(i)(ii)(iii)

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(i)

Let $r$ , $s$ be integers with $r \geq 2$ and $s \geq 2$ . Let $A$ , $B$ be $n \times n$ matrices with $A^r = 0_n$ and $B^s = 0_n$ . Calculate $\mathrm{exp}\left(A \otimes B\right)$ .

(ii)

Let $A$ , $B$ be the basis (given by $n \times n$ matrices) of the non-commutative two-dimensional Lie algebra with $[A,B] = A$ . Calculate $\exp (A \otimes I_n+I_n \otimes A+B \otimes B)$ .

(iii)

Let $X_1$ , $X_2$ , $X_3$ be the basis (given by $n \times n$ matrices) of a Lie algebra with the commutation relations

[X_1,X_2] = 0_n, \qquad [X_1,X_3] = X_1, \qquad [X_2,X_3] = 0_n.

Calculate $\exp (X_1+X_2+X_3)$ and

\exp (X_1 \otimes I_n \otimes I_n+I_n \otimes X_2 \otimes I_n+I_n \otimes I_n \otimes X_3+X_1 \otimes X_2 \otimes X_3).

Problem 9.2.16

Consider the $3 \times 3$ matrix

A = \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \end{pmatrix} = I_3+B, \qquad B = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}.

Calculate $\mathrm{exp}\left(A\right)$ using $e^A = e^{I_3}e^B$ .

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Problem 9.2.17

Let $z \in \mathbb {C}$ . Construct all $2 \times 2$ matrices $A$ and $B$ over $\mathbb {C}$ such that

\mathrm{exp}\left(zA\right)B\mathrm{exp}\left(-zA\right) = e^{-z}B.

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Problem 9.2.18

Let $A$ be an $n \times n$ matrix over $\mathbb {C}$ . Consider the Taylor series

(I_n+A)^{1/2} = I_n+\frac{1}{2}A-\frac{1}{2 \cdot 4}A^2+\frac{1 \cdot 3}{2 \cdot 4 \cdot 6}A^3-\cdots

and

(I_n+A)^{-1/2} = I_n-\frac{1}{2}A+\frac{1 \cdot 3}{2 \cdot 4}A^2-\frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6}A^3+\cdots

What is the condition (the norm) on $A$ such that the Taylor series exist? Can it be applied to the matrix

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

Note that for $n=1$ we have the condition $-1 < A \leq +1$ .

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Problem 9.2.19

Let $U$ be an $n \times n$ unitary matrix. Let $H = U+U^*$ . Calculate $\mathrm{exp}\left(zH\right)$ .

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Problem 9.2.20

Let $A$ , $B$ be $n \times n$ matrices over $\mathbb {C}$ and $\mathrm{exp}\left(A\right)\mathrm{exp}\left(B\right) = \mathrm{exp}\left(C\right)$ . Then the matrix $C$ can be given as an infinite series of commutators of $A$ and $B$ . Let $z \in \mathbb {C}$ . We write

\exp (zA)\exp (zB) = \exp (C(zA,zB))

where

C(zA,zB) = \sum _{j=1}^{\infty } c_j(A,B)z^j.

Show that the expansion up to fourth order is given by

c_1(A,B) = A+B, \qquad c_2(A,B) = \frac{1}{2}[A,B],

c_3(A,B) = \frac{1}{12}[A,[A,B]] - \frac{1}{12}[B,[A,B]], \qquad c_4(A,B) = -\frac{1}{24}[A,[B,[A,B]]].

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Problem 9.2.21

Let $A_1, \ldots , A_m$ be $n \times n$ matrices. Show that

e^{A_1} \cdots e^{A_m} = \exp \left(\frac{1}{2}\sum _{j<k}[A_j,A_k]\right)\exp (A_1+\cdots +A_m)

if the matrices $A_j$ ( $j=1,\ldots ,m$ ) satisfy

[[A_j,A_k],A_\ell ] = 0_n \quad \text{for all } j,k,\ell \in \left\{ 1,\ldots ,m\right\} .

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Problem 9.2.22

Let $A$ , $B$ , $C$ be $n \times n$ positive semidefinite matrices. We define

A \circ B := A^{1/2}BA^{1/2}

where $A^{1/2}$ denotes the unique positive square root of $A$ . Is $A \circ (B \circ C) = (A \circ B) \circ C$ ? Prove or disprove.

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Problem 9.2.23

(i)(ii)

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(i)

Let $\sigma_1$ be the Pauli spin matrix. Show that

\exp \left(-\frac{1}{2}i\pi (\sigma _1-I_2)\right) = \sigma _1.

(ii)

Find all $2 \times 2$ matrices $A$ and $c \in \mathbb {C}$ such that $\exp (c(A-I_2)) = A$ . Note that

\exp (c(A-I_2)) = \exp (cA)\exp (-cI_2) = \exp (cA) \begin{pmatrix} e^{-c} & 0 \\ 0 & e^{-c} \end{pmatrix}.

Problem 9.2.24

Let $A$ , $B$ be real symmetric positive definite matrices. Show that

\frac{1}{\operatorname {det}\left(A\right)}\operatorname {det}\left(A+B\right) \leq \mathrm{exp}\left(\operatorname {tr}(A^{-1}B)\right).

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Problem 9.2.25

Let $\epsilon \in \mathbb {R}$ . Let $I_n - \epsilon A$ be a positive definite matrix. Show that

\exp (\operatorname {tr}(\ln (I_n-\epsilon A))) = \operatorname {det}\left(I_n-\epsilon A\right)

using the identity $\operatorname {det}\left(e^M\right) \equiv \mathrm{exp}\left(\operatorname {tr}(M)\right)$ .

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Problem 9.2.26

Let $\overrightarrow {v}$ be a normalized (column) vector in $\mathbb {C}^{n}$ and let $A$ be an $n \times n$ hermitian matrix. Is $\overrightarrow {v}^*\mathrm{exp}\left(A\right)\overrightarrow {v} \geq \mathrm{exp}\left(\overrightarrow {v}^*A\overrightarrow {v}\right)$ for all normalized $\overrightarrow {v}$ ? Prove or disprove.

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Problem 9.2.27

(i)(ii)(iii)

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(i)

Let $\epsilon \in \mathbb {R}$ . Let $A$ be an $n \times n$ matrix over $\mathbb {C}$ . Find

\lim _{\epsilon \to 0} \frac{\sinh (2\epsilon A)}{\sinh (\epsilon )}.

(ii)

Assume that $A^2 = I_n$ . Calculate

\frac{\sinh (2\epsilon A)}{\sinh (\epsilon )}.

(iii)

Assume that $A^2 = 0_n$ . Calculate

\frac{\sinh (2\epsilon A)}{\sinh (\epsilon )}.

Utilize the expansion

\sinh (2\epsilon A) = 2\epsilon A + \frac{(2\epsilon )^3}{3!}A^3 + \cdots .

Problem 9.2.28

Let $\alpha > 0$ . Consider the $4 \times 4$ matrix

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

Show that $\exp (\pi A(\alpha )) = -I_4$ .

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Problem 9.2.29

Let $V$ be the $2 \times 2$ matrix $V = v_0I_2+v_1\sigma_1+v_2\sigma_2+v_3\sigma_3$ , where $v_0, v_1, v_2, v_3 \in \mathbb {R}$ . Consider the equation

\exp (i\epsilon V) = (I_2-iW)(I_2+iW)^{-1}

where $\epsilon$ is real. Show that $W$ as a function of $V$ is given by

W = -\tan \left(\frac{1}{2}\epsilon (v_0I_2+v_1\sigma _1+v_2\sigma _2+v_3\sigma _3)\right).

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Problem 9.2.30

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

e^{A \otimes I_n+I_n \otimes A}e^{B \otimes I_n+I_n \otimes B} = (e^A \otimes e^B)(e^B \otimes e^B) = (e^Ae^B) \otimes (e^Ae^B).

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Problem 9.2.31

Let $\alpha \in \mathbb {R}$ and $\Pi$ be an $n \times n$ projection matrix. Let $Q = \Pi^{1/2}$ be a square root of $\Pi$ , i.e. $Q^2 = \Pi$ . Show that

U(\alpha ) = \exp (i\alpha \Pi ^{1/2}) = I_n + i\alpha \Pi ^{1/2} + (\cos (\alpha )-1)\Pi + i(\sin (\alpha )-\alpha )\Pi \Pi ^{1/2}.

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Problem 9.2.32

Let $A$ be an $n \times n$ matrix over $\mathbb {R}$ . Then we have the Taylor expansion

\sin (A) := \sum _{k=0}^{\infty } \frac{(-1)^k}{(2k+1)!}A^{2k+1}, \qquad \cos (A) := \sum _{k=0}^{\infty } \frac{(-1)^k}{(2k)!}A^{2k}.

To calculate $\sin (A)$ and $\cos (A)$ from a truncated Taylor series approximation is only worthwhile near the origin. We can use the repeated application of the double angle formula

\cos (2A) \equiv 2\cos ^2(A) - I_n, \qquad \sin (2A) \equiv 2\sin (A)\cos (A).

We can find $\sin (A)$ and $\cos (A)$ of a matrix $A$ from a suitably truncated Taylor series approximates as follows

S_0 = \text{Taylor approximate to } \sin (A/2^k), \qquad C_0 = \text{Taylor approximate to } \cos (A/2^k)

and the recursion

S_j = 2S_{j-1}C_{j-1}, \qquad C_j = 2C_{j-1}^2 - I_n

where $j = 1,2,\ldots$ . Here $k$ is a positive integer chosen so that, say $\left\| A\right\|_{\infty } \approx 2^k$ . Apply this recursion to calculate sine and cosine of the $2 \times 2$ matrix

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

Use $k=2$ .

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Problem 9.2.33

Let $T$ be the $2 \times 2$ matrix

T = \frac{1}{3} \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}.

Calculate $\ln (\operatorname {det}\left(I_2-T\right))$ using the right-hand side of the identity

\ln (\operatorname {det}\left(I_2-T\right)) \equiv -\sum _{k=1}^{\infty } \frac{1}{k}\operatorname {tr}(T^k).

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