Wasin So

Integral circulant graphs

In this note we characterize integral graphs among circulant graphs. It is conjectured that there are exactly 2^@t^(^n^)^-^1 non-isomorphic integral circulant graphs on n vertices, where @t(n) is the number of divisors of n.

On left eigenvalues of a quaternionic matrix

Abstract In this paper, we introduce the concept of left and right eigenvalues for a quaternionic matrix, and investigate their properties, quantities and relationship.

Rank one perturbation and its application to the laplacian spectrum of a graph

Characterization of rank one perturbations of symmetric matrices which change only one eigenvalue are given. Then the result is applied to study how the Laplacian spectrum of a graph changes when adding an edge.

Some explicit formulas for the matrix exponential

Formulas are derived for the exponential of an arbitrary 2*2 matrix in terms of either its eigenvalues or entries. These results are then applied to the second-order mechanical vibration equation with weak or strong damping. Some formulas for the exponential of n*n matrices are given for matrices that satisfy an arbitrary quadratic polynomial. Besides the above-mentioned 2*2 matrices, these results encompass involutory, rank 1, and idempotent matrices. Consideration is then given to n*n matrices that satisfy a special cubic polynomial. These results are applied to the case of a 3*3 skew symmetric matrix whose exponential represents the constant rotation of a rigid body about a fixed axis. >

SINGULAR VALUE INEQUALITY AND GRAPH ENERGY CHANGE

The energy of a graph is the sum of the singular values of its adjacency matrix. A classic inequality for singular values of a matrixsum, including its equality case, is used to study how the energy of a graph changes when edges are removed. One sharp bound and one bound that is never sharp, for the change in graph energy when the edges of a nonsingular induced subgraph are removed, are established. A graph is nonsingular if its adjacency matrixis nonsingular. 1. Singular value inequality for matrix sum. Let X be an n × n complex matrix and denote its singular values by s1(X) ≥ s2(X) ≥ · ·· ≥sn(X) ≥ 0. If X has real eigenvalues only, denote its eigenvalues by λ1(X) ≥ λ2(X) ≥ · ·· ≥λn(X). Define |X| = √ XX ∗ which is positive semi-definite, and note that λi(|X| )= si(X) for all i .W e w riteX ≥ 0t o meanX is positive semi-definite. We are interested in the following singular value inequality for a matrix sum: n � i=1 si(A + B) ≤ n � i=1 si(A )+ n � i=1 si(B)

Quadratic formulas for quaternions

In this paper, we derive explicit formulas for computing the roots of a quaternionic quadratic polynomial.

Commutativity and spectra of Hermitian matrices

Abstract If two Hermitian matrices commute, then the eigenvalues of their sum are just the sums of the eigenvalues of the two matrices in a suitable order. Examples show that the converse is not true in general. In this paper, partial converses are obtained. The technique involves a characterization of the equality cases for Weyls inequalities. Moreover, a new proof on the commutativity of two Hermitian matrices with property L and analogous results for the product of two positive definite Hermitian matrices are included.

Equality cases in matrix exponential inequalities

The Golden–Thompson inequality states that for any Hermitian matrices A and B,

Estimating The Support Of A Scaling Vector

\operatorname{tr} e^A e^B \geqq \operatorname{tr} e^{A + B}

The numerical range of normal matrices with quaternion entries

and the Bernstein inequality states that for any matrix...