ii

We provide an elementary proof for a theorem due to Petz and Réffy which states that for a random n × n unitary matrix with distribution given by the Haar measure on the unitary group U(n), the upper left (or any other) k × k submatrix converges in distribution, after multiplying by a normalization factor √ n and as n → ∞, to a matrix of independent complex Gaussian random variables with mean 0...

We address the local spectral behavior of the random matrix Π1U ⊗kΠ2U ⊗k∗Π1, where U is a Haar distributed unitary matrix of size n×n, the factor k is at most c0 logn for a small constant c0 > 0, and Π1,Π2 are arbitrary projections on l n k 2 of ranks proportional to n. We prove that in this setting the k-fold Kronecker product behaves similarly to the well-studied case when k = 1. AMS Subject ...

Normalized eigenvalue counting measure of the sum of two Hermitian (or real symmetric) matrices An and Bn rotated independently with respect to each other by the random unitary (or orthogonal) Haar distributed matrix Un (i.e. An + U∗ nBnUn) is studied in the limit of large matrix order n. Convergence in probability to a limiting nonrandom measure is established. A functional equation for the St...

We present a source-to-source compiler that processes matrix formulae in the form of Kronecker product factorizations. The Kronecker product notation allows for simple expressions of algorithms such as Walsh-Hadamard, Haar, Slant, Hartley, and FFTs as well as transpositions and wavelet transforms. The compiler is based on a set of term rewriting rules that translate high level matrix descriptio...

In this paper, a new face recognition system based on Wavelet transform (HWT) and Principal Component Analysis (PCA) is presented. The image face is preprocessed and detected. The Haar wavelet is used to form the coefficient matrix for the detected face. The image feature vector is obtained by computing PCA for the coefficient matrix of DWT. A comparison between the proposed recognition system ...

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 linear functions of uniformly distributed random matrices from the classical compact matrix groups. In particular, we provide a new approach for proving the following result of D’Aristotile, Diaconi...

We describe bases of smooth wavelets where the coe cients are obtained by integration against ( nite combinations of) boxcar kernels rather than against traditional smooth wavelets. Bases of this type were rst developed in work of Tchamitchian and of Cohen, Daubechies, and Feauveau. Our approach emphasizes the idea of average-interpolation { synthesizing a smooth function on the line having pre...

A class of parametric transforms that are based on unified representation of transform matrices in the form of sparse matrix products is described. Different families of transforms are defined within the introduced class. All transforms of one family can be computed with fast algorithms similar in structure to each other. In particular, the family of Haar-like transforms consists of discrete or...

In this work we study a version of the general question of how well a Haar distributed orthogonal matrix can be approximated by a random gaussian matrix. Here, we consider a gaussian random matrix Yn of order n and apply to it the Gram-Schmidt orthonormalization procedure by columns to obtain a Haar distributed orthogonal matrix Un. If F m i denotes the vector formed by the first m-coordinates ...