This post is jointly written with Nick Boffi. $ \newcommand{\abs}[1]{| #1 |} \newcommand{\R}{\mathbb{R}} \newcommand{\E}{\mathbb{E}} \newcommand{\T}{\mathsf{T}} \newcommand{\Tr}{\mathrm{Tr}} \newcommand{\ind}{\mathbf{1}} \newcommand{\norm}[1]{\lVert #1 \rVert} $ We give an explicit formula for the volume (w.r.t. the Lebesgue measure in $\mathbb{R}^{n(n+1)/2}$) of real-valued symmetric $n \times n$ matrices with operator norm bounded by one. Specifically, let $S = \{ A \in \mathrm{Sym}_{n} : \norm{A} \leq 1 \}$. We show that $$ \mathrm{Vol}({S}) = \pi^{n(n-1)/4} 2^{n(n+1)/2} \prod_{j=0}^{n-1} \frac{\Gamma(1+j/2)^2}{\Gamma(2 + \frac{n+j-1}{2})} \:, $$ where $S$ is treated as a set in $\R^{n(n+1)/2}$. We thank Liviu Nicolaescu for motivating our approach.

Preliminaries: Gaussian Orthogonal Ensemble

Let $G$ be an $n \times n$ matrix with each entry $G_{ij} \sim N(0, 1)$. Let $A = (G + G^\T)/2$. We say that $A \sim \mathrm{GOE}(n)$.

Lemma: The PDF of $A \sim \mathrm{GOE}(n)$ with respect to the Lebesgue measure on $\R^{n(n+1)/2}$ is: $$ \frac{1}{(2\pi)^{n/2} \pi^{n(n-1)/4}} \exp\left\{ -\frac{1}{2} \Tr(A^2) \right\} \:. $$

Proof: Each entry $A_{ij}$ with $i \leq j$ is independent. Furthermore, $A_{ii} \sim N(0, 1)$ and $A_{ij} \sim N(0, 1/2)$. Hence: $$ \begin{align*} p(A) &= \prod_{i=1}^{n} \frac{1}{(2\pi)^{1/2}} \exp(-a_{ii}/2) \prod_{i < j} \frac{1}{\pi^{1/2}} \exp(-a_{ij}) \\ &= \frac{1}{(2\pi)^{n/2} \pi^{n(n-1)/4}} \exp\left\{- \frac{1}{2}\sum_{i=1}^{n} a_{ii}^2 - \sum_{i < j} a_{ij}^2\right\} \\ &= \frac{1}{(2\pi)^{n/2} \pi^{n(n-1)/4}} \exp(-\Tr(A^2)/2) \:. \end{align*} $$

The follow lemma characterizes the distribution of the eigenvalues of $A \sim \mathrm{GOE}(n)$. As a reference, see Equation 1.4 of Forrester and Warnaar.

Lemma: Let $A \sim \mathrm{GOE}(n)$ and let $\lambda_1, ..., \lambda_n$ denote the eigenvalues of $A$. The PDF of the eigenvalues is: $$ \frac{1}{(2\pi)^{n/2} F_n(1/2)} e^{-\sum_{i=1}^{n} \lambda_i^2/2} \prod_{1 \leq i < j \leq n} \abs{\lambda_i - \lambda_j} \:, $$ where $$ F_n(\gamma) = \prod_{j=1}^{n} \frac{\Gamma(1 + j\gamma)}{\Gamma(1 + \gamma)} \:. $$

Volume Calculation

We can use the GOE density functions to compute the Lebesgue measure of the following set: $$ S := \{ A \in \mathrm{Sym}_{n} : \norm{A} \leq 1 \} \:, $$ where we treat the set as a subset of $\R^{n(n+1)/2}$. We do this as follows. First, we observe that: $$ \E_{A \sim \mathrm{GOE}(n)}[ \ind\{ \norm{A} \leq 1\} \exp(\Tr(A^2)/2) ] = \frac{1}{(2\pi)^{n/2} \pi^{n(n-1)/4}} \int_{\norm{A} \leq 1} d\mu = \frac{\mathrm{Vol}({S})}{(2\pi)^{n/2} \pi^{n(n-1)/4} } \:. $$ On the other hand, letting $\mathrm{eigs}(n)$ denote the distribution over the eigenvalues of matrices from $\mathrm{GOE}(n)$, $$ \begin{align*} \E_{A \sim \mathrm{GOE}(n)}[ \ind\{ \norm{A} \leq 1\} \exp(\Tr(A^2)/2) ] &= \E_{\lambda_i \sim \mathrm{eigs}(n)}\left[ \prod_{i=1}^{n} \ind\{ \abs{\lambda_i} \leq 1 \} \exp\left( \sum_{i=1}^{n} \lambda_i^2/2 \right)\right] \\ &= \frac{1}{(2\pi)^{n/2} F_n(1/2)} \int_{-1}^{1} ... \int_{-1}^{1} \prod_{1 \leq i < j \leq n} \abs{\lambda_i - \lambda_j} \: d\lambda_1 \:...\: d\lambda_n \\ &= \frac{2^{n(n+1)/2}}{(2\pi)^{n/2} F_n(1/2)}\int_{0}^{1} ... \int_{0}^{1} \prod_{1 \leq i < j \leq n} \abs{\lambda_i - \lambda_j} \: d\lambda_1 \:...\: d\lambda_n \:. \end{align*} $$ The integral is a Selberg integral which equals: $$ \int_{0}^{1} ... \int_{0}^{1} \prod_{1 \leq i < j \leq n} \abs{\lambda_i - \lambda_j} \: d\lambda_1 \:...\: d\lambda_n = \prod_{j=0}^{n-1}\frac{\Gamma(1 + \frac{j}{2})^2\Gamma(1 + \frac{j+1}{2})}{\Gamma(2 + \frac{n+j-1}{2})\Gamma(\frac{3}{2})} \:. $$ Therefore: $$ \frac{\mathrm{Vol}({S})}{(2\pi)^{n/2} \pi^{n(n-1)/4} } = \frac{2^{n(n+1)/2}}{(2\pi)^{n/2} F_n(1/2)}\prod_{j=0}^{n-1}\frac{\Gamma(1 + \frac{j}{2})^2\Gamma(1 + \frac{j+1}{2})}{\Gamma(2 + \frac{n+j-1}{2})\Gamma(\frac{3}{2})} \:. $$ Solving for $\mathrm{Vol}({S})$, $$ \mathrm{Vol}({S}) = \pi^{n(n-1)/4} 2^{n(n+1)/2} \prod_{j=0}^{n-1} \frac{\Gamma(1+j/2)^2}{\Gamma(2 + \frac{n+j-1}{2})} \:. $$ This is the desired result.