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The determinant of a complex matrix and Gershgorin circles

Author: Bünger Florian
Rump Siegfried M.
Publication venue: Wyoming Scholars Repository
Publication date: 2019
Field of study

Each 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

Effects on the distance Laplacian spectrum of graphs with clusters by adding edges

Author: Díaz Roberto C.
Rojo Oscar
Publication venue: Wyoming Scholars Repository
Publication date: 2019
Field of study

All 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| \ge 2

of independent vertices sharing the same set

S

|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

\partial-|C|+2

is a distance Laplacian eigenvalue of

G(H)

with multiplicity

|C|-1

, where

\partial

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

The sum of the first two largest signless Laplacian eigenvalues of trees and unicyclic graphs

Author: Du Zhibin
Publication venue: Wyoming Scholars Repository
Publication date: 2019
Field of study

Let

G

be a graph on

n

vertices with

e(G)

edges. The sum of eigenvalues of graphs has been receiving a lot of attention these years. Let

S_2 (G)

be the sum of the first two largest signless Laplacian eigenvalues of

G

, and define

f(G) = e (G) +3 - S_2 (G)

. Oliveira et al. (2015) conjectured that

f(G) \geqslant f(U_{n})

with equality if and only if

G \cong U_n

, where

U_n

is the

n

-vertex unicyclic graph obtained by attaching

n-3

pendent vertices to a vertex of a triangle. In this paper, it is proved that

S_2(G) \u3c e(G) + 3 -\frac{2}{n}

when

G

is a tree, or a unicyclic graph whose unique cycle is not a triangle. As a consequence, it is deduced that the conjecture proposed by Oliveira et al. is true for trees and unicyclic graphs whose unique cycle is not a triangle

The $A_{\alpha}$ - spectrum of graph product

Author: Li Shuchao
Wang Shujing
Publication venue: Wyoming Scholars Repository
Publication date: 2019
Field of study

Let

A(G)

and

D(G)

denote the adjacency matrix and the diagonal matrix of vertex degrees of

G

, respectively. Define

A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)

for any real

\alpha\in [0,1]

. The collection of eigenvalues of

A_{\alpha}(G)

together with multiplicities is called the

A_{\alpha}

-\emph{spectrum} of

G

. Let

G\square H

G[H]

G\times H

and

G\oplus H

be the Cartesian product, lexicographic product, directed product and strong product of graphs

G

and

H

, respectively. In this paper, a complete characterization of the

A_{\alpha}

-spectrum of

G\square H

for arbitrary graphs

G

and

H

, and

G[H]

for arbitrary graph

G

and regular graph

H

is given. Furthermore,

A_{\alpha}

-spectrum of the generalized lexicographic product

G[H_1,H_2,\ldots,H_n]

for

n

-vertex graph

G

and regular graphs

H_i

\u27s is considered. At last, the spectral radii of

A_{\alpha}(G\times H)

and

A_{\alpha}(G\oplus H)

for arbitrary graph

G

and regular graph

H

are given

Parametrized solutions $X$ of the system $AXA = AY A$ and $A^k Y AX = XAY A^k$

Author: Ferreyra David
Lattanzi Marina
Levis Fabián
Thome Néstor
Publication venue: Wyoming Scholars Repository
Publication date: 2019
Field of study

Let

A

and

E

n \times 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

A^k E A X = X A E A^k

, 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

Weak Log-majorization of Unital Trace-preserving Completely Positive Maps

Author: Lau Pan Shun
Tam Tin-Yau
Publication venue: Wyoming Scholars Repository
Publication date: 2019
Field of study

Let \Phi:\bM_n\to\bM_n be a unital trace preserving completely positive map and A\in\bM_n be a positive definite matrix. Weak log-majorization and weak majorization between

\Phi(A)

and

A

are studied. Determinantal inequalities between

\Phi(A)

and

A

are obtained as a consequence. By considering special classes of unital trace preserving completely positive map, some known matrix inequalities such as Fischer\u27s inequality are rediscovered. An affirmative answer to a question of Tam and Zhang in 2019 is given

Left in the Dust: Wyoming’s Instream Flow Laws from a Mountain West Perspective

Author: Connell Allison E.
Publication venue: Wyoming Scholars Repository
Publication date: 2019
Field of study

Jordan Triple Product Homomorphisms on Triangular Matrices to and from Dimension One

Author: Kokol Bukovsek Damjana
Mojskerc Blaz
Publication venue: Wyoming Scholars Repository
Publication date: 2019
Field of study

A map

\Phi

is a Jordan triple product (JTP for short) homomorphism whenever

\Phi(A B A)= \Phi(A) \Phi(B) \Phi(A)

for all

A,B

. We study JTP homomorphisms on the set of upper triangular matrices

\mathcal{T}_n(\mathbb{F})

, where \Ff is the field of real or complex numbers. We characterize JTP homomorphisms

\Phi: \mathcal{T}_n(\mathbb{C}) \to \mathbb{C}

and JTP homomorphisms

\Phi: \mathbb{F} \to \mathcal{T}_n(\mathbb{F})

. In the latter case we consider continuous maps and the implications of omitting the assumption of continuity

Cone-constrained rational eigenvalue problems

Author: Seeger Alberto
Publication venue: Wyoming Scholars Repository
Publication date: 2019
Field of study

This work deals with the eigenvalue analysis of a rational matrix-valued function subject to complementarity constraints induced by a polyhedral cone

K

. The eigenvalue problem under consideration has the general structure

\left(\sum_{k=0}^d \lambda^k A_k + \sum_{k =1}^m \frac{p_k(\lambda)}{q_k(\lambda)} \,B_k\right) x = y , \quad K\ni x \perp y\in K^\ast,

where

K^\ast

denotes the dual cone of

K

. The unconstrained version of this problem has been discussed in [Y.F. Su and Z.J. Bai. Solving rational eigenvalue problems via linearization. \emph{SIAM J. Matrix Anal. Appl.}, 32:201--216, 2011.] with special emphasis on the implementation of linearization-based methods. The cone-constrained case can be handled by combining Su and Bai\u27s linearization approach and the so-called facial reduction technique. In essence, this technique consists in solving one unconstrained rational eigenvalue problem for each face of the polyhedral cone

K

Non-sparse Companion Matrices

Author: Deaett Louis
Fischer Jonathan
Garnett Colin
Vander Meulen Kevin N
Publication venue: Wyoming Scholars Repository
Publication date: 2019
Field of study

Given a polynomial

p(z)

, a companion matrix can be thought of as a simple template for placing the coefficients of

p(z)

in a matrix such that the characteristic polynomial is

p(z)

. The Frobenius companion and the more recently-discovered Fiedler companion matrices are examples. Both the Frobenius and Fiedler companion matrices have the maximum possible number of zero entries, and in that sense are sparse. In this paper, companion matrices are explored that are not sparse. Some constructions of non-sparse companion matrices are provided, and properties that all companion matrices must exhibit are given. For example, it is shown that every companion matrix realization is non-derogatory. Bounds on the minimum number of zeros that must appear in a companion matrix, are also given

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