On spectral and numerical properties of random butterfly matrices

Spectral and numerical properties of classes of random orthogonal butterfly matrices, as introduced by Parker (1995), are discussed, including the uniformity of eigenvalue distributions. These matrices are important because the matrix-vector product with an $N$-dimensional vector can be performed in $O(N \log N)$ operations. And in the simplest situation, these random matrices coincide with Haar measure on a subgroup of the orthogonal group. We discuss other implications in the context of randomized linear algebra.