Fomin and Zelevinsky [FZ1] have defined cluster algebras, and developed an interesting and influential theory about this class of algebras. We deal here with a special case of cluster algebras. Let B be an n×n integral skew symmetric matrix, or equivalently a finite quiver QB with n vertices and no loops or oriented cycles of length two. Let u = {u1, . . . , un} be a transcendence basis for F =...

For a simple digraph G of order n with vertex set {v1, v2, . . . , vn}, let d+i and d − i denote the out-degree and in-degree of a vertex vi in G, respectively. Let D (G) = diag(d+1 , d + 2 , . . . , d + n ) and D−(G) = diag(d1 , d − 2 , . . . , d − n ). In this paper we introduce S̃L(G) = D̃(G)−S(G) to be a new kind of skew Laplacian matrix of G, where D̃(G) = D+(G)−D−(G) and S(G) is the skew-adj...

A class of symmetric static output feedback controllers are known to robustly stabilize symmetric collocated second-order linear time invariant systems having positive definite or positive semi-definite coefficient matrices. This paper extends the result to the asymmetric systems which include skew-symmetric gyroscopic terms. We first obtain the condition for static output feedback controllers ...

Given a graph G, let G be an oriented graph of G with the orientation σ and skewadjacency matrix S(G). Then the spectrum of S(G) consisting of all the eigenvalues of S(G) is called the skew-spectrum of G, denoted by Sp(G). The skew energy of the oriented graph G, denoted by ES(G), is defined as the sum of the norms of all the eigenvalues of S(G). In this paper, we give orientations of the Krone...

The minimum (symmetric) rank of a simple graph G over a field F is the smallest possible rank among all symmetric matrices over F whose ijth entry (for i 6= j) is nonzero whenever {i, j} is an edge in G and is zero otherwise. The problem of determining minimum (symmetric) rank has been studied extensively. We define the minimum skew rank of a simple graph G to be the smallest possible rank amon...

We study the problem of feedback control for skew-symmetric and skewHamiltonian transfer functions using skew-symmetric controllers. This extends work of Helmke, et al., who studied static symmetric feedback control of symmetric and Hamiltonian linear systems. We identify spaces of linear systems with symmetry as natural subvarieties of the moduli space of rational curves in a Grassmannian, giv...

An algorithm is presented for online learning of rotations. The proposed algorithm involves matrix exponentiated gradient updates and is motivated by the von Neumann divergence. The multiplicative updates are exponentiated skew-symmetric matrices which comprise the Lie algebra of the rotation group. The orthonormality and unit determinant of the matrix parameter are preserved using matrix logar...

Let A be a skew matrix of order n over an ordered field. There Is a finite class of skew matrices A such that XA = Y and XA = Y have the same solution sets, where x = y. and y = x for some Indices 1 (perhaps none) and x = x , y = y for the remaining 1 . We show that for each Index h, 1 < h < n, there exists an A such that ä > 0 for all J . A proof of von Neumann's Mlnlmax Theorem for symmetric ...

We consider the direct solution of sparse skew symmetric matrices. We see that the pivoting strategies are similar, but simpler, to those used in the factorization of sparse symmetric indefinite matrices, and we briefly describe the algorithms used in a forthcoming direct code based on multifrontal techniques for the factorization of real skew symmetric matrices. We show how this factorization ...