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Invertible bases and root vectors for analytic matrix-valued functions
No full textWe revisit the concept of a minimal basis through the lens of the theory of modules over a commutative ring . We first review the conditions for the existence of a basis for submodules of where is a Bézout domain. Then, we define the concept of invertible basis of a submodule of and when is an elementary divisor domain, we link it to the Main Theorem of G. D. Forney Jr. [SIAM J. Control, 13:493-520, 1975]. Over an elementary divisor domain, the submodules admitting an invertible basis are precisely the free pure submodules of . As an application, we let be either a connected compact set or a connected open set, and we specialize to , the ring of functions that are analytic on . We show that, for any matrix , is a free -module and admits an invertible basis, or equivalently a basis that is full rank upon evaluation at any . Finally, given , we use invertible bases to define and study maximal sets of root vectors at for . This in particular allows us to define eigenvectors also for analytic matrices that do not have full column rank
Matrices having nonzero outer inverses
No full textIt is well known that every nonzero von Neumann regular -matrix over an arbitrary ring has a nonzero outer inverse -matrix in the sense that . Generalizing previous work on von Neumann regular matrices, the matrices having nonzero outer inverses over semiperfect rings are characterized as the matrices having some entry outside the Jacobson radical of . Such matrices over finite semiperfect rings and finite commutative rings are counted, and several applications are given
Minimizing the least eigenvalue of unbalanced signed unicyclic graphs with given girth or pendant vertices
No full textA signed graph consists of an underlying graph with a sign function . Let be the adjacency matrix of . Let be the spectrum of the signed graph , where is the least eigenvalue of . Let denote the set of all the unbalanced signed unicyclic graphs with order , girth and pendant vertices, let denote the set of all the unbalanced signed unicyclic graphs with vertices and pendant vertices, and let denote the set of all the unbalanced signed unicyclic graphs with order and girth . Obviously, and . In this paper, we determine the signed unicyclic graphs whose least eigenvalues are minimal among all the graphs in , and , respectively
Locating Eigenvalues of Symmetric Matrices - A Survey
No full textWe survey algorithms for locating eigenvalues of symmetric matrices taking advantage of the underlying graph. We present applications in spectral graph theory
Cycle products and efficient vectors in reciprocal matrices
No full textWe focus on the relationship between Hamiltonian cycle products and efficient vectors for a reciprocal matrix , to more deeply understand the latter. This facilitates a new description of the set of efficient vectors (as a union of convex subsets), greater understanding of convexity within this set and of order reversals in efficient vectors. A straightforward description of all efficient vectors for an -by-, column perturbed consistent matrix is given; it is the union of at most convex sets
Coalescing sets preserving cospectrality of graphs arising from block similarity matrices
No full textCoalescing involves gluing one or more rooted graphs onto another graph. Under specific conditions, it is possible to start with cospectral graphs that are coalesced in similar ways that will result in new cospectral graphs. We present a sufficient condition for this based on the block structure of similarity matrices, possibly with additional constraints depending on which type of matrix is being considered. The matrices considered in this paper include the adjacency, Laplacian, signless Laplacian, distance, and generalized distance matrix
Characterization of invariant subspaces for a nilpotent linear operator that admit complementary invariant subspaces
No full textThe aim of this work is to solve the problem of determining the necessary and sufficient conditions for a vector subspace invariant by a nilpotent endomorphism to admit a complementary invariant subspace for the same linear operator. As applications, we offer results about Jordan bases associated with nilpotent linear maps and reflexive generalized inverses of finite potent endomorphisms and square matrices
Girls’ Class and Character in Contemporary YA Fiction
No full textTwo contemporary works of young adult fiction that appeared on the New York City 365 Book List are examined in this paper as examples of the socioeconomic diversity that the creators of the list intended to include. Nic Stone’s Jackpot (2019) and Ibi Zoboi’s Pride (2018) both depict female high schoolers—Rico and Zuri—who grapple with their own identity-building as they demonstrate awareness of their socioeconomic situations, both of working-class backgrounds, and how their personal contexts contribute to the future opportunities available to them. Ultimately, even as the novels are propelled forward with romantic relationships with male teens of higher socioeconomic statuses, with greater access to power and privilege, both protagonists ultimately develop agency in their lives and powerfully negotiate their futures, especially through their understandings and analyses of class and its connection to their identities and relationships
