Math Problems Vault

Problem 25.2.1

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

\exp (\beta (A+B)) \equiv \exp (\beta A) \left(I_{n} + \int _{0}^{\beta } d\epsilon \, e^{-\epsilon A} B e^{\epsilon (A+B)}\right).

Multiply both sides from the left with $\exp (-\beta A)$ and differentiate with respect to $\beta$ .

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Problem 25.2.2

Let $\alpha \in \mathbb {R}$ . Consider an $n \times n$ matrix $A(\alpha )$ , where the entries of $A$ depends smoothly on $\alpha$ . Then one has the identity

\frac{d}{d\alpha } e^{A(\alpha )} \equiv \int _{0}^{1} e^{(1-s)A(\alpha )} \frac{dA(\alpha )}{d\alpha } e^{sA(\alpha )} \, ds \equiv \int _{0}^{1} e^{sA(\alpha )} \frac{dA(\alpha )}{d\alpha } e^{(1-s)A(\alpha )} \, ds.

Let $n = 2$ and

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

Calculate the right-hand sides of the identities.

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Problem 25.2.3

Let $A$ , $B$ be positive definite matrices. Then we have the integral representation ( $x \geq 0$ )

\ln (A+xB) - \ln (A) \equiv \int _{0}^{\infty } (A+uI_{n})^{-1} xB (A+xB+uI_{n})^{-1} \, du.

Let

A = \begin{pmatrix} 2 & -1 \\ -1 & 1 \end{pmatrix}, \qquad B = \begin{pmatrix} 1 & 1 \\ 1 & 2 \end{pmatrix}.

Calculate the left- and right-hand side of the integral representation.

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Problem 25.2.4

Consider the $2 \times 2$ matrices

A(t) = \begin{pmatrix} \cos (t) & -\sin (t) \\ \sin (t) & \cos (t) \end{pmatrix}, \qquad B(t) = \int _{0}^{t} A(s) \, ds.

Find the commutator $[A(t), B(t)]$ . Discuss. What is the condition such that $[A(t), B(t)] = 0_{2}$ ?

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Problem 25.2.5

Let $P_{j}$ ( $j = 0, 1, 2, \ldots$ ) be the Legendre polynomials

P_{0}(x) = 1, \qquad P_{1}(x) = x, \qquad P_{2}(x) = \frac{1}{2}(3x^{2}-1), \qquad \ldots .

Show that the infinite dimensional matrix $A = (a_{jk})$

a_{jk} := \int _{-1}^{+1} P_{j}(x) \frac{dP_{k}(x)}{dx} \, dx

where $j, k = 0, 1, \ldots$ is given by

A = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & \cdots \\ 2 & 0 & 0 & 0 & 0 & 0 & 0 & \cdots \\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & \cdots \\ 2 & 0 & 2 & 0 & 0 & 0 & 0 & \cdots \\ 0 & 2 & 0 & 2 & 0 & 0 & 0 & \cdots \\ 2 & 0 & 2 & 0 & 2 & 0 & 0 & \cdots \\ 0 & 2 & 0 & 2 & 0 & 2 & 0 & \cdots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{pmatrix}.

Note that the matrix is not symmetric. The matrix $A$ can be considered as a linear operator in the Hilbert space $\ell_{2}(\mathbb {N}^{0})$ . Is $\left\| A\right\| < \infty$ ?

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Problem 25.2.6

(i)(ii)

Let $A$ , $B$ , $C$ be positive definite $n \times n$ matrices. Then (Lieb inequality)

\operatorname {tr}(e^{\ln (A)-\ln (B)+\ln (C)}) \leq \operatorname {tr}\int _{0}^{\infty } A(B+uI_{n})^{-1} C(B+uI_{n})^{-1} \, du.

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(i)

Let

A = \begin{pmatrix} 2 & -1 \\ -1 & 1 \end{pmatrix}, \qquad B = \begin{pmatrix} 1 & 1 \\ 1 & 2 \end{pmatrix}, \qquad C = \begin{pmatrix} 4 & 2 \\ 2 & 5 \end{pmatrix}.

Calculate the left-hand side and right-hand side of the inequality.

(ii)

Show that a sufficient condition such that one has an equality is that $A$ , $B$ , $C$ commute. However this condition is not necessary.