Let $G = (V, E)$ be a simple graph. Denote by $D(G)$ the diagonal matrix $diag(d_1,cdots,d_n)$, where $d_i$ is the degree of vertex $i$  and  $A(G)$ the adjacency matrix of $G$. The  signless Laplacianmatrix of $G$ is $Q(G) = D(G) + A(G)$ and the $k-$th signless Laplacian spectral moment of  graph $G$ is defined as $T_k(G)=sum_{i=1}^{n}q_i^{k}$, $kgeqslant 0$, where $q_1$,$q_2$, $cdots$, $q_n$ ...

For a square (0, 1,−1) sign pattern matrix S, denote the qualitative class of S by Q(S). In this paper, we investigate the relationship between sign patterns and matrices that diagonalise an irreducible nonnegative matrix. We explicitly describe the sign patterns S such that every matrix in Q(S) diagonalises some irreducible nonnegative matrix. Further, we characterise the sign patterns S such ...

A scaled version of the lower and the upper triangular factors of the inverse of the Vandermonde matrix is given. Two applications of the q-Pascal matrix resulting from the factorization of the Vandermonde matrix at the q-integer nodes are introduced. c © 2007 Elsevier Ltd. All rights reserved.

A q-analogue of Sudarshan’s diagonal representation of the Quantum Mechanical density matrix is obtained using q-boson coherent states. Earlier result of Mehta and Sudarshan on the self reproducing property of ρ(z, z) is also generalized and a self-consistent self-reproducing kernel K̃(z, z) is constructed. e-mail addresses: [email protected] ; [email protected] 1 A diagonal representati...

R. C. Buck fl] has shown that if a regular matrix sums every subsequence of a sequence x, then x is convergent. I. J. Maddox [4] improved Buck's theorem by showing that if a non-Schur matrix sums every subsequence of a sequence x, then x is convergent. Actually Maddox proved a stronger result: If x is divergent and A sums every subsequence of x, then A is a Schur matrix, i.e., It should be rema...

Let $G$ be a strongly connected digraph with vertex set $\{v_1, v_2, \dots, v_n\}$. Denote by $D_{ij}$ the distance between vertices $v_i$ and $v_j$ in $G$. Two variant versions of matrix were proposed Yan Yeh (Adv. Appl. Math.), Bapat et al. (Linear Algebra Appl.) independently, one is $q$-distance matrix, other exponential matrix. Given nonzero indeterminate $q$, $\mathscr{D}_G=(\mathscr{D}_{...

The QR-algorithm is a renowned method for computing all eigenvalues of an arbitrary matrix. A preliminary unitary similarity transformation to Hessenberg form is indispensible for keeping the computational complexity of the QRalgorithm applied on the resulting Hessenberg matrix under control. The unitary factor Q in the QR-factorization of the Hessenberg matrix H = QR is composed of n − 1 rotat...

We solve optimization problems on the ranks and inertias of the quadratic Hermitian matrix function Q − XPX∗ subject to a consistent system of matrix equations AX = C and XB = D. As applications, we derive necessary and sufficient conditions for the solvability to the systems of matrix equations and matrix inequalities AX = C,XB = D, and XPX∗ = (>, <, ≥, ≤)Q in the Löwner partial ordering to be...

We show that two important quantities from two disparate areas of complexity theory — Strassen’s exponent of matrix multiplication ω and Grothendieck’s constant KG — are intimately related. They are different measures of size for the same underlying object — the matrix multiplication tensor, i.e., the 3-tensor or bilinear operator μl,m,n : Fl×m × Fm×n → Fl×n, (A,B) 7→ AB defined by matrix-matri...

For a prime power q ≡ 1 (mod v), the q × q cyclotomic matrix, whose entries are the discrete logarithmsmodulo v of the entries in the addition table of Fq , has been shown using character theoretic arguments to produce an ε-biased array, provided that q is large enough as a function of v and ε. A suitable choice of ε ensures that the array is a covering array of strength t when q > t2v4t . On t...