Singular value of partially prescribed matrices, ETFs, etc...

In this talk I will present two linear algebra problems, motivated by the engineering practice. In the first problem, we study singular values of a matrix whose one entry varies while all other entries are prescribed. In particular, we find the possible p-th singular value of such a matrix, and we define explicitly the unknown entry such that the completed matrix has the minimal possible p-th singular value. This in turn determines possible p-th singular value of a matrix under rank one perturbation. Moreover, we determine the possible value of p-th singular value of a partially prescribed matrix whose set of unknown entries has a form of a Young diagram, i.e. if the entry (i,j) is known then all the entries (x,y), with x greater than or equal to i and y greater than or equal to j, are known. In particular, we give a fast algorithm for defining the completion that minimizes the p-th singular value of such matrix. The second problem consist in finding the set of N lines in the d-dimensional real or complex space, such that pairwise angles are as large as possible. This corresponds to the packing problems in the projective spaces. We describe some partial results, and explain relations with various fields including number theory, quantum measurements and antena communications. At the end, I will present some open problems related to both parts of the lectures.