# Linear functions on the classical matrix groups

@article{Meckes2005LinearFO,
title={Linear functions on the classical matrix groups},
author={Elizabeth S. Meckes},
journal={Transactions of the American Mathematical Society},
year={2005},
volume={360},
pages={5355-5366}
}
• Elizabeth S. Meckes
• Published 20 September 2005
• Mathematics
• Transactions of the American Mathematical Society
Let M be a random matrix in the orthogonal group On, distributed according to Haar measure, and let A be a fixed n x n matrix over R such that Tr(AA t ) = n. Then the total variation distance of the random variable Tr(AM) to a standard normal random variable is bounded by 2√3 n-1 and this rate is sharp up to the constant. Analogous results are obtained for M a random unitary matrix and A a fixed n x n matrix over C. The proofs are applications of a new abstract normal approximation theorem…

### High-Dimensional Random Matrices from the Classical Matrix Groups, and Generalized Hypergeometric Functions of Matrix Argument

Results from the theory of the generalized hypergeometric functions of matrix argument, and the related zonal polynomials, are used to develop a new approach to study the asymptotic distributions of

### Rate of convergence of linear functions on the unitary group

• Mathematics
• 2010
We study the rate of convergence to a normal random variable of the real and imaginary parts of , where U is an N × N random unitary matrix and AN is a deterministic complex matrix. We show that the

• Mathematics
• 2019

### Central Limit Theorems for the Brownian motion on large unitary groups

In this paper, we are concerned with the large N limit of linear combinations of the entries of a Brownian motion on the group of N by N unitary matrices. We prove that the process of such a linear

### The Distribution of Permutation Matrix Entries Under Randomized Basis

We study the distribution of entries of a random permutation matrix under a “randomized basis,” i.e., we conjugate the random permutation matrix by an independent random orthogonal matrix drawn from

### Lattice Walks in a Weyl Chamber and Truncated Random Matrices

Let u(d, n) denote the number of permuations in the symmetric group Sn with no increasing subsequence of length greater than d. u(d, n) may alternatively be interpreted as the number of closed

### Convergence of moments of twisted COE matrices

• Mathematics
• 2020
We investigate eigenvalue moments of matrices from Circular Orthogonal Ensemble multiplicatively perturbed by a permutation matrix. More precisely we investigate variance of the sum of the

### Random orthogonal matrices and the Cayley transform

• Mathematics, Computer Science
Bernoulli
• 2020
An asymptotic independent normal approximation is established for the distribution of the Euclidean parameters which corresponds to the uniform distribution on the Stiefel manifold and contributes to the growing literature on normal approximations to the entries of random orthogonal matrices or transformations thereof.

## References

SHOWING 1-10 OF 22 REFERENCES

### ON RANDOM MATRICES FROM THE COMPACT CLASSICAL GROUPS

If M is a matrix taken randomly with respect to normalized Haar measure on U(n), O(n) or Sp(n), then the real and imaginary parts of the random variables Tr(Mk), k > 1, converge to independent normal

### Maxima of entries of Haar distributed matrices

Abstract.Let Γn=(γij) be an n×n random matrix such that its distribution is the normalized Haar measure on the orthogonal group O(n). Let also Wn:=max1≤i,j≤n|γij|. We obtain the limiting distribution

### Brownian motion and the classical groups

• Mathematics
• 2002
Let G be chosen from the orthogonal group On according to Haar measure, and let A be an ? ? ? real matrix with non-random entries satisfying TrAA1 = n. We show that TrAT converges in distribution to

### Finite de Finetti theorems in linear models and multivariate analysis

• Mathematics
• 1992
Let Xl,-.. , Xk be a sequence of random vectors. We give symmetry conditions on the joint distribution which imply that it is well approximated by a mixture of normal distributions. Examples include

### How many entries of a typical orthogonal matrix can be approximated by independent normals

We solve an open problem of Diaconis that asks what are the largest orders of p n and q n such that Z n , the p n x q n upper left block of a random matrix r n which is uniformly distributed on the

### The Central Limit Problem for Random Vectors with Symmetries

• Mathematics
• 2007
Abstract Motivated by the central limit problem for convex bodies, we study normal approximation of linear functionals of high-dimensional random vectors with various types of symmetries. In

### Invariant Integration over the Unitary Group

• Mathematics
• 2003
Integrals for the product of unitary-matrix elements over the U(n) group will be discussed. A group-theoretical formula is available to convert them into a multiple sum, but unfortunately the sums

### A Combinatorial Central Limit Theorem

Let (Y n1,…,Y nn be a random vector which takes on the n! permutations of (1,…, n) with equal probabilities. Let c n(i,j), i,j = 1, …, n, be n real numbers. Sufficient conditions for the asymptotic

### PATTERNS IN EIGENVALUES: THE 70TH JOSIAH WILLARD GIBBS LECTURE

Typical large unitary matrices show remarkable patterns in their eigenvalue distribution. These same patterns appear in telephone encryption, the zeros of Riemann’s zeta function, a variety of

### The semicircle law, free random variables, and entropy

• Mathematics
• 2000
Overview Probability laws and noncommutative random variables The free relation Analytic function theory and infinitely divisible laws Random matrices and asymptotically free relation Large