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On m-th roots of complex matrices

Author: Liu Heguo
Zhao Jing
Publication venue: International Linear Algebra Society
Publication date: 20/08/2022
Field of study

For an

n\times n

matrix

M

\sigma(M)

denotes the set of all different eigenvalues of

M

. In this paper, we will prove two results on the

m

-th

(m\geq2)

roots of a matrix

A

. Firstly, let

X

be an

m

-th root of

A

. Then

X

can be expressed as a polynomial in

A

if and only if rank

X^2

= rank

X

and

|\sigma(X)|=|\sigma(A)|

. Secondly, let

X

and

Y

be two

m

-th roots of

A

. If both

X

and

Y

can be expressed as polynomials in

A

, then

X=Y

if and only if

\sigma(X)=\sigma(Y)

Kronecker products of Perron similarities

Author: Dockter Janelle M.
Paparella Pietro
Perry Robert L.
Ta Jonathan D.
Publication venue: International Linear Algebra Society
Publication date: 14/02/2022
Field of study

An invertible matrix is called a Perron similarity if one of its columns and the corresponding row of its inverse are both nonnegative or both nonpositive. Such matrices are of relevance and import in the study of the nonnegative inverse eigenvalue problem. In this work, Kronecker products of Perron similarities are examined and used to construct ideal Perron similarities all of whose rows are extremal

Compatibility and companions for Leonard pairs

Author: Nomura Kazumasa
Terwilliger Paul
Publication venue: International Linear Algebra Society
Publication date: 20/08/2022
Field of study

In this paper, we introduce the concepts of compatibility and companion for Leonard pairs. These concepts are roughly described as follows. Let

\mathbb{F}

denote a field, and let

V

denote a vector space over

\mathbb{F}

with finite positive dimension.A Leonard pair on

V

is an ordered pair of diagonalizable

\mathbb{F}

-linear maps

A : V \to V

and

A^* : V \to V

that each act in an irreducible tridiagonal fashion on an eigenbasis for the other one. Leonard pairs

A,A^*

and

B,B^*

V

are said to be compatible whenever

A^* = B^*

and

[A,A^*] = [B,B^*]

, where

[r,s] = r s - s r

. For a Leonard pair

A,A^*

V

, by a companion of

A,A^*

we mean an

\mathbb{F}

-linear map

K: V \to V

such that

K

is a polynomial in

A^*

and

A-K, A^*

is a Leonard pair on

V

. The concepts of compatibility and companion are related as follows. For compatible Leonard pairs

A,A^*

and

B,B^*

V

, define

K = A-B

. Then

K

is a companion of

A,A^*

. For a Leonard pair

A,A^*

V

and a companion

K

A,A^*

,define

B = A-K

and

B^* = A^*

. Then

B,B^*

is a Leonard pair on

V

that is compatible with

A,A^*

. Let

A,A^*

denote a Leonard pair on

V

. We find all the Leonard pairs

B, B^*

V

that are compatible with

A,A^*

.For each solution

B, B^*

, we describe the corresponding companion

K = A-B

Bounds via spectral radius-preserving row sum expansions

Author: Stover Joseph P.
Publication venue: International Linear Algebra Society
Publication date: 22/08/2022
Field of study

We show a simple method for constructing larger dimension nonnegative matrices with somewhat arbitrary entries which can be irreducible or reducible but preserving the spectral radius via row sum expansions. This yields a sufficient criteria for two square nonnegative matrices of arbitrary dimension to have the same spectral radius, a way to compare spectral radii of two arbitrary square nonnegative matrices, and a way to derive new upper and lower bounds on the spectral radius which give the standard row sum bounds as a special case

Group inverses of matrices of directed trees

Author: Nandi R.
Sivakumar K. C.
Publication venue: International Linear Algebra Society
Publication date: 07/10/2022
Field of study

A new class of directed trees is introduced. A formula for the group inverse of the matrices associated with any tree belonging to this class is obtained. This answers affirmatively, a conjecture of Catral et al., for this new class

Programming Bee-Bots to Draw Geometric Shapes

Author: Alqahtani Muteb
Hall Jacob
Publication venue: University of Wyoming Libraries
Publication date: 17/12/2022
Field of study

This article describes an activity that integrates Bee-Bot robots, to engage first-grade students in creating different geometric shapes. In this activity, students work collaboratively to program Bee-Bots to move on a specific path with two markers attached to the Bee-Bots to create a geometric shape on a sheet of paper. After exploring the robots, first-grade students are asked to create geometric shapes such as squares, rectangles, and circles. Students write and test their programs, then observe the drawing to debug their program. This allows students to attend to the number of sides and vertices and the relations among them to complete the task.

Considering Structure in Online Learning

Author: Shepherd Craig E.
Publication venue: University of Wyoming Libraries
Publication date: 17/12/2022
Field of study

This week-long, asynchronous online module introduces the theory of transactional distance, particularly the component associated with structure, to graduate-level instructional design students enrolled in a seminar course about online learning. Learners were also introduced to publicly available Quality Matters rubrics and other readings associated with the effective design of online instruction. To demonstrate their understanding of the topic, learners imported skeletal learning resources into a course shell and developed them further in terms of sequence, clarity, organization, and purpose based on assigned readings and discussions. Because dialogue and accessibility concepts were covered more fully in later modules, these topics were minimized during this module. Topics: distance education, online learning, learning management system, course design, structure, clarity, organization. Time:  One week was allocated for this lesson

A Personalized Learning Choice Board for Blended Learning Teacher Preparation

Author: Short Cecil R.
Arnesen Karen T
Publication venue: University of Wyoming Libraries
Publication date: 17/12/2022
Field of study

This unit allows preservice teachers at the end of a one-credit hour, K-12, online and blended teaching course to further explore blended teaching competencies that they feel are weak or are interested in. The unit is meant to be completed independently with weekly asynchronous check-ins to ensure progress. The unit includes a choice board of activities focused on the four different blended teaching competency areas covered in prior units. Students select and complete three activities from the choice board. Each activity provides an opportunity to reflect on the activity and on experiences with personalized learning

Some additional notes on the spectra of non-negative symmetric 5 x 5 matrices

Author: Loewy Raphael
Publication venue: International Linear Algebra Society
Publication date: 09/01/2021
Field of study

The Symmetric Non-negative Inverse Eigenvalue Problem (SNIEP) asks when is a list

\sigma = \left( \lambda_{1}, \lambda_{2}, \dots, \lambda_{n} \right)

of real, monotonically decreasing numbers, the spectrum of an

n \times n

, symmetric, non-negative matrix

A

. In that case, we say

\sigma

is realizable and

A

is a realizing matrix. Here, we consider the case

n=5

, the lowest value of

n

for which the problem is unsolved. Let

s_{1}(\sigma) = \sum_{i=1}^5 \lambda_{i}

and

s_{3}(\sigma) = \sum_{i=1}^5 {\lambda_{i}}^3

. It is known that to complete the solution for

n=5

, it remains to consider the case \lambda_{3} > s_{1}(\sigma), so let

y=\lambda_{3}- s_{1}(\sigma)

and assume

y \geq 0

. We prove that if

\sigma

is realizable, then

s_{3}(\sigma) \geq s_{1}(\sigma)^3+6s_{1}(\sigma)y(s_{1}(\sigma)+y)

. This strengthens the inequality

s_{3}(\sigma) \geq s_{1}(\sigma)^3

obtained by Loewy and Spector, which in turn strengthens the inequality

25s_{3}(\sigma) \geq s_{1}(\sigma)^3

, one of the Johnson--Loewy--London inequalities. As an application of the new inequality, we show that certain lists previously unknown as far as their realizability is concerned are not realizable

Nonsparse companion Hessenberg matrices

Author: Borobia Alberto
Canogar Roberto
Publication venue: International Linear Algebra Society
Publication date: 05/03/2021
Field of study

In recent years, there has been a growing interest in companion matrices. Sparse companion matrices are well known: every sparse companion matrix is equivalent to a Hessenberg matrix of a particular simple type. Recently, Deaett et al. [Electron. J. Linear Algebra, 35:223--247, 2019] started the systematic study of nonsparse companion matrices. They proved that every nonsparse companion matrix is nonderogatory, although not necessarily equivalent to a Hessenberg matrix. In this paper, the nonsparse companion matrices which are unit Hessenberg are described. In a companion matrix, the variables are the coordinates of the characteristic polynomial with respect to the monomial basis. A PB-companion matrix is a generalization, in the sense that the variables are the coordinates of the characteristic polynomial with respect to a general polynomial basis. The literature provides examples with Newton basis, Chebyshev basis, and other general orthogonal bases. Here, the PB-companion matrices which are unit Hessenberg are also described

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