Linear Algebra of Magic Squares

A magic square M is an n-by-n array of numbers whose rows, columns, and the two diagonals sum to µ called the magic sum. If all the diagonals including broken diagonals sum to µ then the magic square is said to be pandiagonal. A regular magic square satisfies the condition that. If the entries of M are the entries symmetrically placed with respect to the center sum to 2µ n and M is said to be a classical magic square. In this paper, we find vector space dimension of regular and pandiagonal magic squares. We give a simpler proof of the known result that even order regular magic squares are singular. We present matrix theoretic constructions that produce odd order regular magic squares that are singular and nonsingular. integers 1 through n2 the magic sum µ is n(n2 +1)

On structural decompositions of finite frames

A frame in an n-dimensional Hilbert space Hn is a possibly redundant collection of vectors {f[subscript i]}[subscript i∈I] that span the space. A tight frame is a generalization of an orthonormal basis. A frame {f[subscript i]}[subscript i∈I] is said to be scalable if there exist nonnegative scalars {c[subscript i]}[subscript i∈I] such that {c[subscript i]f[subscript i]}[subscript i∈I] is a tight frame. In this paper we study the combinatorial structure of frames and their decomposition into tight or scalable subsets by using partially-ordered sets (posets). We define the factor poset of a frame {f[subscript i]}[subscript i∈I] to be a collection of subsets of I ordered by inclusion so that nonempty J⊆I is in the factor poset iff {f[subscript j]}[subscript j∈J] is a tight frame for Hn. We study various properties of factor posets and address the inverse factor poset problem, which inquires when there exists a frame whose factor poset is some given poset P. We then turn our attention to scalable frames and present partial results regarding when a frame can be scaled to have a given factor poset; in doing so we present a bridge between erasure resilience (as studied via prime tight frames) and scalability.National Science Foundation (U.S.). Research Experience for Undergraduates (Program) (Grant DMS 11-56890