Rank structured matrices: theory, algorithms and applications

In numerical linear algebra much attention has been paid to matrices that are sparse, i.e., containing a lot of zeros. For example, to compute the eigenvalues of a general dense symmetric matrix, this matrix is first reduced to a similar tridiagonal one using an orthogonal similarity transformation. The subsequent QR-algorithm performed on this n×n tridiagonal matrix, takes the sparse structure of this matrix into account resulting in O(n) flops per iteration compared to O(n3) flops for a general dense matrix. Much less attention was given to so-called rank structured matrices although they share similar theoretical and computational properties. As an example one can look at the inverse of a tridiagonal matrix with nonzero subdiagonal elements. This inverse is generically a dense matrix. However, all submatrices taken from the lower (upper) triangular part of this matrix all have rank one. Such a matrix is called a semiseparable matrix. The aim of this series of lectures is to gain insight into this class of matrices and some of its generalizations. These matrices are called rank structured matrices. Rank structured matrices can be loosely defined as matrices that have one or more submatrices of low rank. Similar to sparse matrices where the sparsity pattern can be structured or unstructured, rank structured matrices can be considered having a structured or unstructured pattern of the low rank matrices. We will focus in these lectures on the rank structured matrices with a specific pattern in the low rank submatrices. As time permits, one or more of the following topics will be covered: • In the literature different names and slightly different definitions are given for rank structured matrices. We will study these different definitions. Besides the different definitions also several different representations can be used. We will discuss the advantages and disadvantages of these representations.