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The - spectrum of graph product
No full textLet and denote the adjacency matrix and the diagonal matrix of vertex degrees of , respectively. Define for any real . The collection of eigenvalues of together with multiplicities is called the -\emph{spectrum} of . Let , , and be the Cartesian product, lexicographic product, directed product and strong product of graphs and , respectively. In this paper, a complete characterization of the -spectrum of for arbitrary graphs and , and for arbitrary graph and regular graph is given. Furthermore, -spectrum of the generalized lexicographic product for -vertex graph and regular graphs 's is considered. At last, the spectral radii of and for arbitrary graph and regular graph are given
Inequalities for permanents and permanental minors of row substochastic matrices
No full textIn this paper, some inequalities for permanents and permanental minors of row substochastic matrices are proved. The convexity of the permanent function on the interval between the identity matrix and an arbitrary row substochastic matrix is also proved. In addition, a conjecture about the permanent and permanental minors of square row substochastic matrices with fixed row and column sums is formulated
Parametrized solutions of the system and
No full textLet A and E be n × n given complex matrices. This paper provides a necessary and sufficient condition for the solvability to the matrix equation system given by AXA = AEA and AkEAX = XAEAk, for k being the index of A. In addition, its general solution is derived in terms of a G-Drazin inverse of A. As consequences, new representations are obtained for the set of all G-Drazin inverses; some interesting applications are also derived to show the importance of the obtained formulas
The determinant of a complex matrix and Gershgorin circles
No full textEach connected component of the Gershgorin circles of a matrix contains exactly as many eigenvalues as circles are involved. Thus, the Minkowski (set) product of all circles contains the determinant if all circles are disjoint. In [S.M. Rump. Bounds for the determinant by Gershgorin circles. Linear Algebra and its Applications, 563:215--219, 2019.], it was proved that statement to be true for real matrices whose circles need not to be disjoint. Moreover, it was asked whether the statement remains true for complex matrices. This note answers that in the affirmative. As a by-product, a parameterization of the outer loop of a Cartesian oval without case distinction is derived
The cone of Z-transformations on Lorentz cone
No full textIn this paper, the structural properties of the cone of Z-transformations on the Lorentz cone are described in terms of the semidefinite cone and copositive/completely positive cones induced by the Lorentz cone and its boundary. In particular, its dual is described as a slice of the semidefinite cone as well as a slice of the completely positive cone of the Lorentz cone. This provides an example of an instance where a conic linear program on a completely positive cone is reduced to a problem on the semidefinite cone
On a Refined Operator Version of Young's Inequality and Its Reverse
No full textIn this note, some refinements of Young's inequality and its reverse for positive numbers are proved, and using these inequalities, some operator versions and Hilbert-Schmidt norm versions for matrices of these inequalities are obtained
An Eigenvalue Approach For Estimating The Generalized Cross Validation Function For Correlated Matrices
No full textThis work proposes a fast estimate for the generalized cross-validation function when the design matrix of an experiment has correlated columns. The eigenvalue structure of this matrix is used to derive probability bounds satisfied by an appropriate index of proximity, which provides a simple and accurate estimate for the numerator of the generalized cross-validation function. The denominator of the function is evaluated by an analytical formula. Several simulation tests performed in statistical models having correlated design matrix with intercept confirm the reliability of the proposed probabilistic bounds and indicate the applicability of the proposed estimate for these models
Effects on the distance Laplacian spectrum of graphs with clusters by adding edges
No full textAll graphs considered are simple and undirected. A cluster in a graph is a pair of vertex subsets (C, S), where C is a maximal set of cardinality |C| ≥ 2 of independent vertices sharing the same set S of |S| neighbors. Let G be a connected graph on n vertices with a cluster (C, S) and H be a graph of order |C|. Let G(H) be the connected graph obtained from G and H when the edges of H are added to the edges of G by identifying the vertices of H with the vertices in C. It is proved that G and G(H) have in common n −|C| + 1 distance Laplacian eigenvalues, and the matrix having these common eigenvalues is given, if H is the complete graph on |C| vertices then ∂ −|C| + 2 is a distance Laplacian eigenvalue of G(H) with multiplicity|C| − 1, where ∂ is the transmission in G of the vertices in C. Furthermore, it is shown that if G is a graph of diameter at least 3, then the distance Laplacian spectral radii of G and G(H) are equal, and if G is a graph of diameter 2, then conditions for the equality of these spectral radii are established. Finally, the results are extended to graphs with two or more disjoint clusters
Washing lines, whinberries and reworking ‘waste ground’: Women's affective practices and a haunting within the haunting of the UK coalfields
No full textThis article reflects on a series of ‘Ghost lab’ events (Bright 2019) with local people where creative memory work – stimulated by songs, films, and readings from a pack of what we have called a ‘Community Tarot’ cards (our main focus here) – was used to register aspects of what, following Gordon (2008), we are calling a ‘social haunting’ of former coal-mining communities in the north of England and the valley communities of south Wales. The events were part of a joint 2018-19 research project called Song lines on the road – Life lines on the move! (On the Road for short) that sought to share two independent strands of longitudinal, co-produced, arts-based research in which we have developed approaches aimed at amplifying how living knowledge flows on in communities even when the shocks and intensities of lived experience defy articulation and representation. During the last decade or so both of us have worked with artists to co-produce research projects that enable young people and marginalised adults to communicate with and challenge authority by drawing on the affective power of art. Independently of each other until now, we have both been using creative/affective methodologies to understand how classed and gendered circuits of affect both reproduce and reconfigure vernacular bonds of solidarity and practices of wellbeing in multiple impoverished coalfield communities
Solving the Sylvester Equation AX-XB=C when
No full textThe method for solving the Sylvester equation in complex matrix case, when , by using Jordan normal form is given. Also, the approach via Schur decomposition is presented