Search Results

Indeed, if $A$ is non-Hermitian you cannot use the inverse $(A-z)^{-1}$ to study eigenvalues near a complex number $z$. To apply resolvent techniques to a non-Hermitian matrix $A$ you need to first sy …

Indeed, the distribution function of the eigenphases of a random matrix in $\operatorname{SO}(n)$ has a peak at 0 and at $\pm\pi$. It only becomes uniform for large $n$. The joint distribution functio …

The Fourier transform of the marginal distribution of a single eigenvalue in the GUE is known, $$f_{{\rm GUE}(d)}(k,0,0,\ldots,0)=e^{-\tfrac{1}{2}k^2/d}\sum_{j=0}^{d-1}(-1)^jk^{2j}\frac{(d-1)(d-2)\cdo …

A proof is on page 80-81 and 90 of Forrester, the probability distribution function of $W=X^\top X$ is $\propto e^{-\tfrac{1}{2}\operatorname{tr}W}(\operatorname{det}W)^{(n-m-1)/2}$, for an $n\times m …

Use the Euler angle parameterisation of the rotation matrix, $$Z=\begin{pmatrix} R(\alpha)&0\\ 0&1\end{pmatrix} \begin{pmatrix} 1&0\\ 0&R(\theta)\end{pmatrix} \begin{pmatrix} R(\alpha')&0\\ 0&1\end{pm …

The Poisson kernel for the orthogonal group was calculated by Benjamin Béri in Generalization of the Poisson kernel to the superconducting random-matrix ensembles. This is in the context where $X$ is …

You want to think of the Haar measure $d\mu(U)$ as a way of measuring uniformity in the group $U(N)$ of unitary $N\times N$ matrices. To form your intuition, consider $N=1$. You then have $U=e^{i\phi} …

The $\sqrt{n}$ factors can be absorbed in the $X$ matrices, I can omit them. Consider a Hermitian matrix $X$ and a Hermitian perturbation $\delta X$. Then, for $f:\mathbb{R}\rightarrow\mathbb{R}$ one …

Since $$Z=X^\top (XX^\top + \mathrm{Id})^{-1} X=(Y + \mathrm{Id})^{-1} Y,$$ with $Y=X^\top X$, we can find the eigenvalue density of $Z$ from the eigenvalue density of $Y$, which for large matrix dime …

Here are some applications of free probability of random matrices: Neural networks: The asymptotic freeness assumption plays a fundamental role in the study of the propagation of spectral distributio …

There is a substantial literature; for an introduction, you could start with the chapter by Keating and Snaith in the Handbook of Random Matrix Theory. For a more specialized overview, take a look at …

This follows from the fact that $\mathbb{E}[A^\dagger A]=d I$ (with $A^\dagger$ the conjugate transpose of $A$ and $I$ the $d\times d$ identity matrix). Hence $$\mathbb{E}[\|C_n\|_F^2]=\operatorname{t …

I use results from Forrester's article (written for a rank-one perturbation of a matrix from the GOE, but more generally valid for an isotropic random-matrix ensemble). A. Largest eigenvalue Equation …

Since $O$ is orthogonal, $O^\top=O^{-1}$ commutes with $O$, hence the eigenvalues $\mu_n$ of $O+O^\top$ are related to the eigenvalues $e^{i\phi_n}$ of $O$ by $\mu_n=2\cos\phi_n$. The spectral density …

This alternative proof of the Keating-Snaith formula avoids the Selberg integral and may be what you after: The characteristic polynomial of a random unitary matrix: a probabilistic approach, by Paul …