In statistics, the Wishart distribution is a generalization of the gamma distribution to multiple dimensions. It is named in honor of John Wishart, who first formulated the distribution in 1928. Other names include Wishart ensemble (in random matrix theory, probability distributions over matrices are usually called "ensembles"), or Wishart–Laguerre ensemble (since its eigenvalue distribution involve Laguerre polynomials), or LOE, LUE, LSE (in analogy with GOE, GUE, GSE).

It is a family of probability distributions defined over symmetric, positive-definite random matrices (i.e. matrix-valued random variables). These distributions are of great importance in the estimation of covariance matrices in multivariate statistics. In Bayesian statistics, the Wishart distribution is the conjugate prior of the inverse covariance-matrix of a multivariate-normal random vector.

Suppose G is a p × n matrix, each column of which is independently drawn from a p-variate normal distribution with zero mean:

It means ${\displaystyle g_{i}=(g_{i,1},\dots ,g_{i,p})^{T}\ {\overset {iid}{\sim }}\ {\mathcal {N}}_{p}(0,V)\ \forall i\in \{1,\dots ,n\}}$

Then the Wishart distribution is the probability distribution of the p × p random matrix

known as the scatter matrix. One indicates that S has that probability distribution by writing

The positive integer n is the number of degrees of freedom. Sometimes this is written W(V, p, n). For n ≥ p the matrix S is invertible with probability 1 if V is invertible.

If p = V = 1 then this distribution is a chi-squared distribution with n degrees of freedom.