Estimating the condition numbers of random structured matrices is a well known challenge (cf. [SST06]), linked to the design of efficient randomized matrix algorithms in [PGMQ], [PIMR10], [PQ10], [PQ12], [PQZa], [PQa], [PQZb], [PQZC], [PY09]. We deduce such estimates for Gaussian random Toeplitz and circulant matrices. The former estimates can be surprising because the condition numbers grow ex...

In present paper, we investigate 4 problems. Firstly, it is known that, a matrix is MDS if and only if all sub-matrices of this matrix of degree from 1 to n are full rank. In this paper, we propose a theorem that an orthogonal matrix is MDS if and only if all sub-matrices of this orthogonal matrix of degree from 1 to bn2 c are full rank. With this theorem, calculation of constructing orthogonal...

A multilevel circulant is defined as a graph whose adjacency matrix has a certain block decomposition into circulant matrices. A general algebraic method for finding the eigenvectors and the eigenvalues of multilevel circulants is given. Several classes of graphs, including regular polyhedra, suns, and cylinders can be analyzed using this scheme.