Integration and Matrices

Consider the $2 \times 2$ matrix over $\mathbb {R}$

Find

Then calculate the commutator $[F(s), G(t)]$.

Consider the analytic function $f : \mathbb {C} \rightarrow \mathbb {C}$, $f(z) = z^{2}$ and the $2 \times 2$ matrix

Calculate $f(A)$ applying the Cauchy integral theorem.

Let $A$ be an $n \times n$ matrix over $\mathbb {C}$. If $\lambda$ is not an eigenvalue of $A$, then the matrix $(A - \lambda I_{n})$ has an inverse, namely the resolvent

Let $\lambda_{j}$ be the eigenvalues of $A$. For $\left|\lambda \right| \geq a$, where $a$ is any positive constant greater than all the $\left|\lambda_{j}\right|$ the resolvent can be expanded as

Calculate

Let $\mathbb {C}^{n \times N}$ be the vector space of all $n \times N$ complex matrices. Let $Z \in \mathbb {C}^{n \times N}$. Then $Z^{*} \equiv \overline{Z}^{T}$, where $T$ denotes transpose. One defines a Gaussian measure $\mu$ on $\mathbb {C}^{n \times N}$ by

where $dZ$ denotes the Lebesgue measure on $\mathbb {C}^{n \times N}$. The Fock space $\mathcal{F}(\mathbb {C}^{n \times N})$ consists of all entire functions on $\mathbb {C}^{n \times N}$ which are square integrable with respect to the Gaussian measure $d\mu (Z)$. With the scalar product

one has a Hilbert space. Show that this Hilbert space has a reproducing kernel $K$. This means a continuous function $K(Z, Z') : \mathbb {C}^{n \times N} \times \mathbb {C}^{n \times N} \rightarrow \mathbb {C}$ such that

for all $Z \in \mathbb {C}^{n \times N}$ and $f \in \mathcal{F}(\mathbb {C}^{n \times N})$.

Let $A$ be an $n \times n$ positive definite matrix over $\mathbb {R}$, i.e.$\overrightarrow {x}^{T}A\overrightarrow {x} > 0$ for all $\overrightarrow {x} \in \mathbb {R}^{n}$. Calculate

Let $V$ be an $N \times N$ unitary matrix, i.e.$VV^{*} = I_{N}$. The eigenvalues of $V$ lie on the unit circle; that is, they may be expressed in the form $\exp (i\theta_{n})$, $\theta_{n} \in \mathbb {R}$. A function $f(V) = f(\theta_{1}, \ldots , \theta_{N})$ is called a class function if $f$ is symmetric in all its variables. Weyl gave an explicit formula for averaging class functions over the circular unitary ensemble

Thus we integrate the function $f(V)$ over $U(N)$ by parametrizing the group by the $\theta_{i}$ and using Weyl’s formula to convert the integral into an $N$-fold integral over the $\theta_{i}$. By definition the Haar measure $dV$ is invariant under $V \rightarrow \widetilde{U}V\widetilde{U}^{*}$, where $\widetilde{U}$ is any $N \times N$ unitary matrix. The matrix $V$ can always be diagonalized by a unitary matrix, i.e.

where $W$ is an $N \times N$ unitary matrix. Thus the integral over $V$ can be written as an integral over the matrix elements of $W$ and the eigenphases $\theta_{n}$. Since the measure is invariant under unitary transformations, the integral over the matrix elements of $U$ can be evaluated straightforwardly, leaving the integral over the eigenphases. Show that for $f$ a class function we have

Let $A$ be an $n \times n$ positive definite matrix over $\mathbb {R}$. Let $\overrightarrow {q}$ and $\overrightarrow {J}$ be column vectors in $\mathbb {R}^{n}$. Calculate

Note that

Let $A$ be an $n \times n$ positive definite matrix over $\mathbb {R}$, i.e.all the eigenvalues, which are real, are positive. We also have $A^{T} = A$. Consider the analytic function $f : \mathbb {R}^{n} \rightarrow \mathbb {R}$

Calculate the Fourier transform of $f$. The Fourier transform is defined by

where $\overrightarrow {k}\cdot \overrightarrow {x} \equiv \overrightarrow {k}^{T}\overrightarrow {x} \equiv k_{1}x_{1} + \cdots + k_{n}x_{n}$ and $d\overrightarrow {x} = dx_{1}\ldots dx_{n}$. The inverse Fourier transform is given by

where $d\overrightarrow {k} = dk_{1}\ldots dk_{n}$. With $a > 0$ we have

Let $A$ be an $n \times n$ matrix. Let $\omega , \mu \in \mathbb {R}$. Assume that

and $\mu > \omega$. Then we have

Calculate the left- and right-hand side of (1) for the matrix $A = \sigma_{1}$.

Consider the differentiable manifold

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

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

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

Let $n = 2$ and

Calculate the right-hand sides of the identities.

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

Let

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

Consider the $2 \times 2$ matrices

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

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

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

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

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 $\left\| A\right\| < \infty$?

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