Random matrix

A random matrix is a matrix whose entries are random variables. Its eigenvalues form a random configuration, and its empirical spectral measure records their distribution with equal mass on each eigenvalue. Entry assumptions and normalization determine the spectral regime.

In the unscaled convention with diagonal variance one, a Gaussian unitary ensemble matrix $H$ is a Hermitian matrix with independent real diagonal entries $N(0,1)$ and upper off-diagonal entries $(A_{ij}+i{B}_{ij})/\sqrt{2}$, where all the $A_{ij},B_{ij}$ and diagonal entries are independent and have the standard normal distribution. Its density on the real coordinates of Hermitian matrices is proportional to $\exp [-\mathrm{tr}(H^2)/2]$. Its exchangeably labelled real eigenvalues have symmetric joint probability density function proportional to ${\prod }_{i<j}({\lambda }_i-{\lambda }_j{)}^2{e}^{-{\sum }_i{\lambda }_i^2/2}$. The squared Vandermonde determinant suppresses coincident eigenvalues; orthogonal polynomial projection kernels express the correlation functions of a point process as determinants.

In the unscaled variance-one convention, a Gaussian orthogonal ensemble matrix $M$ is a real symmetric matrix whose upper-triangular entries are independent, with $M_{ij}\sim N(0,1)$ for $i<j$ and $M_{ii}\sim N(0,2)$. Its joint probability density function relative to the upper-triangular coordinates is proportional to $\exp [-\mathrm{tr}(M^2)/4]$. The scaled convention uses $M/\sqrt{n}$; the scaling must be stated. If $X$ has independent standard normal entries, $(X+X^T)/\sqrt{2}$ realizes the unscaled convention. Expanding in the entries of $X$ gives the covariance identity $\mathbb{E}[M_{ij}{M}_{kl}]={\delta }_{ik}{\delta }_{jl}+{\delta }_{il}{\delta }_{jk}$, with Kronecker deltas.

A standard real Gaussian random matrix has independent and identically distributed standard normal entries. Each column is an orthogonally invariant random vector: its probability density function is proportional to $\exp (-\Vert z{\Vert }_2^2/2)$. Consequently multiplying on either side by a fixed orthogonal matrix preserves the joint distribution. Its singular values are random; the expressions $\sqrt{m}\pm \sqrt{n}$ for an $m\times n$ matrix describe useful spectral scales, not deterministic endpoints that hold for every realization.

For integer $\nu$, the sum ${\sum }_{j=1}^{\nu }{X}_j{X}_j^T$ of independent $N_p(0,V)$ vectors has a Wishart distribution. Its positive-definite density extends to real $\nu >p-1$ and is proportional to $|W{|}^{(\nu -p-1)/2}\exp [-\mathrm{tr}(V^{-1}W)/2]$. It is a conjugate prior for the precision matrix of a multivariate normal distribution. Its mean is $\nu V$.

The WinBUGS declaration dwish(R, nu) uses $R=V^{-1}$; the density convention is documented in the NIMBLE BUGS-language manual. For $\nu =p=3$ it is proper, while the expected inverse matrix does not exist: that expectation requires $\nu >p+1$. Specifying a Wishart prior on precision is therefore a substantive covariance assumption.

If $W\sim W_p(\nu ,R^{-1})$ has the Wishart distribution, then $W^{-1}$ has the inverse-Wishart distribution. For $\nu >p+1$, its expected value is $R/(\nu -p-1)$; below this threshold the matrix mean does not exist. It commonly models a covariance matrix, but couples the prior behavior of variances and correlations.

For an $N$-by-$N$ Hermitian matrix, the empirical spectral measure puts mass $1/N$ at each eigenvalue, counted with multiplicity. It is a probability measure. Integrating a test function against it gives the average of that function evaluated at the eigenvalues.

A real Wigner matrix is a real symmetric matrix whose upper off-diagonal entries are independent centered copies of a fixed random variable, scaled to have variance of order $1/N$. Diagonal conventions may vary and should be specified separately. A common version has variance-one unscaled off-diagonal entries and either zero diagonal entries or independent diagonal entries with a bounded second moment.

Truncate the unscaled entries of a Wigner matrix at a fixed threshold and subtract their expected values. The difference between original and truncated entries is $N^{-1/2}(Z_C-\mathbb{E}{Z}_C)$ with $Z_C=Y{1}_{|Y|\ge C}$, so its variance is bounded by $N^{-1}\mathbb{E}[Y^2{1}_{|Y|\ge C}]$. The centering term cannot be discarded for an asymmetric entry law.

For a fixed mean-zero variance-one real entry law, centered truncation of a Wigner matrix satisfies $\mathbb{E}[N^{-1}\mathrm{Tr}(X-\hat{X}{)}^2]\le t(C)$, where $t(C)=\mathbb{E}[Y^2{1}_{|Y|\ge C}]\to 0$. The spectral Lipschitz bound from Frobenius distance and Markov inequality give probability at most $t(C)/{\epsilon }^2$ for a spectral test-function discrepancy exceeding $\epsilon$. Thus $t(C)<{\epsilon }^3$ works uniformly in matrix size. A varying entry-law family requires uniform second-moment tail decay instead.

A complex Wishart matrix is the Gram matrix of an $n$-by-$m$ matrix of independent centered complex Gaussian random variables with an indicated variance convention. It is a positive semidefinite matrix. In the standard convention with joint entry density proportional to $e^{-\mathrm{Tr}{X}^{\ast }X}$ and $n\ge m$, its eigenvalue density contains the weight $x^{n-m}{e}^{-x}$ and a squared Vandermonde determinant.

The Laguerre unitary ensemble is the positive-eigenvalue ensemble with joint density proportional to ${\prod }_{i<j}(x_j-x_i{)}^2{\prod }_i{x}_i^a{e}^{-x_i}$, for $a>-1$. Integer $a=n-m\ge 0$ is realized by a complex Wishart matrix. The Laguerre projection kernel expresses its correlation functions of a point process as determinants.