Given a real, symmetric matrix S, we define the slice FS through S as being the connected component containing S of two orbits under conjugation: the first by the orthogonal group, and the second by the upper triangular group. We describe some classical constructions in eigenvalue computations and integrable systems which keep slices invariant — their properties are clarified by the concept. We...

In [1] and [2] a new family of companion forms associated to a regular polynomial matrix T (s) has been presented, using products of permutations of n elementary matrices, generalizing similar results presented in [3] where the scalar case was considered. In this paper, extending this “permuted factors” approach, we present a broader family of companion like linearizations, using products of up...

A graph is called symmetric if its full automorphism group acts transitively on the set of arcs. The Cayley graph $Gamma=Cay(G,S)$ on group $G$ is said to be normal symmetric if $N_A(R(G))=R(G)rtimes Aut(G,S)$ acts transitively on the set of arcs of $Gamma$. In this paper, we classify all connected tetravalent normal symmetric Cayley graphs of order $p^2q$ where $p>q$ are prime numbers.

Abstract. Let A be an n by n symmetric matrix with real entries. Using the l1-norm for vectors and letting S 1 = {x ∈ R|||x||1 = 1, x ≥ 0}, the matrix A is said to be interior if the quadratic form x Ax achieves its minimum on S 1 in the interior. Necessary and sufficient conditions are provided for a matrix to be interior. A copositive matrix is referred to as being exceptional if it is not th...

Let A be an n by n symmetric matrix with real entries. Using the l1-norm for vectors and letting S 1 = {x ∈ Rn|||x||1 = 1, x ≥ 0}, the matrix A is said to be interior if the quadratic form xT Ax achieves its minimum on S 1 in the interior. Necessary and sufficient conditions are provided for a matrix to be interior. A copositive matrix is referred to as being exceptional if it is not the sum of...

Let (X,S) be an association scheme where X is a finite set and S is a partition of X × X. We say that (X,S) is symmetric if σs is symmetric for each s ∈ S where σs is the adjacency matrix of s, and integral if ⋃ s∈S ev(σs) ⊆ Z where ev(σs) is the set of all eigenvalues of σs. For an association scheme (X,T ) we say that (X,T ) is a fusion of (X,S) if each element of T is a union of elements of ...

We describe some of the determinantal ideals attached to symmetric, exterior and tensor powers of a matrix. The methods employed use elements of Zariski's theory of complete ideals and of representation theory. Let R be a commutative ring. The determinantal ideals attached to matrices with entries in R play ubiquitous roles in the study of the syzygies of R{modules. In this note, we describe so...

We describe some of the determinantal ideals attached to symmetric, exterior and tensor powers of a matrix. The methods employed use elements of Zariski’s theory of complete ideals and of representation theory. Let R be a commutative ring. The determinantal ideals attached to matrices with entries in R play ubiquitous roles in the study of the syzygies of R–modules. In this note, we describe so...