JAVA matrix algorithms

Sang-Gu Lee

Sungkyunkwan Univ., Korea

sglee@math.skku.ac.kr

2002. 1. 14. Postech

Abstract

We shall show Multimedia Web Contents of Linear Algebra which was developed under Korea Research Foundation Project # 2000-048-D0001.

** (with
YoonMee Ham, Rich Wellman, Han-Guk Seol)**

This contains several JAVA Applets, Flash tools and Animations that enables the visualization of concepts in Linear Algebra.

Examples :

(i) Macromedia Flash : SWF 1(LT), SWF 2(LT), SWF
3(LT), SWF 4(LT), SWF 5(LT), **SWF
6(LT-Movie) with ,**

(ii) JAVA: <Interactive JAVA Applets>

(iii) Intoductions (mpeg), Cover

(iv) Book of Definitions and Theorems

etc

<Background>

(1) KMS Newsletter V.70 P. 23, 2000

"Disscussion on Math Major Education in Korea"

(2) KMS and Com2Mac, July, 2000

"Symposium on Diversity of Math Dept. in Korea"

Some Conclusion:

(i) Mathematics for the Math. Major and Mathematics

- Intensity and identity

(ii) Mathematics classes as a service course

- Development of New courses, New contents or New way of teaching for Engineering and Science major

(iii) Mathematics for ordinary peoples

- Diverse Approach

(3) KMS Newsletter V.79 P.18, 2001

Prof. Dalwon Park of Kongju Univ.

"Web Based Instruction on Calculus"

<Main>

We are going to introduce

**"Multimedia Web Contents of Linear Algebra****" **

which was developed under Korea Research Foundation

Project # 2000-048-D0001.

- Character -

(1) Multimedia

(2) Increase Interaction

(3) JAVA

(4) Macromedia Flash

(5) Animation

-Visualizeation of Concepts on Web

Why Linear Algebra?

(i) Calculus, Engineering Math,

(ii) **LA,** DE, Discrete Math, NA,

<Motivation>

(1) Internet Environment in Korea :

(i) ADSL Access from Apt,

(ii) Every class room was connected by Jan. 2001.

(2) Cyber Universities in Korea :

(i) Nine 4 year Universities was opened in 2001

(ii) 7 more Universities will open in 2002

Engineering major such as Telecommunication and Information Technology is popular : Math courses are needed.

The way to Design :

(1) Objective

(2) Sample Exam, Quizzes

(3) 14 weeks Lectures

(4) Mid Term, Comprehensive Inclass Final

(5) Discussion, Q & A

(6) WBI Capability

(7) Personalized contents : Background, Appl., links

(8) Multimedia

(9) Interactive

(10) Computational ability but Software Independant (JAVA Applet).

(11) Applications: Mathematical Modelling

**So we wanted to meet the request of KRF's 4 criteria;
**

**⼑ Increase effectiveness of Education
**

**⼑ Edu. Info. Tech for Lifelong Education in 21st Century
**

**⼑ Student oriented interactive Edu. via Internet
**

**⼑ Genuine Multimedia Contents by hightech media
**

Our "Multimedia Web Contents of Linear Algebra"

Each week (14 weeks) :

System of Linear Equations and Matrices, Matrix operations and Guassian Eliminations, Inverse, Cryptography and Matrix, Determinants, Cofactor expansion, Applications of Determinant, Vector Spaces and Subspaces, Basis and Dimension of Solution space, Gram-Schmidt O.N. process, Linear Transformations
SWF, SWF II , Matrices of L.T., Analytic Geometry of Euclidean Space
**SWF
(V)**, Orthogonal Projections and Bases, Similarity, Eigenvalues and Eigenvectors, Diagonalization of Symmetric Matrices,

**SWF
6(LT-Movie) with ,**

Understanding of Concepts, Computational skill, Modeling ability

******* ****Matrices of L.T.**
* Interactive JAVA Applets

Optional sections:

Population Dynamics, Linear Diff. Equations, Complex inner product space, Special matrices, Jordan canonical form, Quadratic form and diagonalization, Mathematical Modelling, Least square curve fitting etc,

**SWF
7(Volume)**, SWF 8(Trace), SWF
9(Inverse), SWF 10(Transpose), SWF
11(Product), **SWF 12(Det),****
****SWF 13(Movie1)**** **

Reading materials:

History, Software(MATHEMATICA, Excell, HLINPRAC, Mathrix) for LA, Mathematics in Press, Mathematicians, ATLAST Project, Visualization of Concepts by MATLAB.

<An example of Flash Animation (REF)>

SWF 1, SWF 2, SWF
3, SWF 4(LT), SWF 5(LT), **SWF
6(diagonalization),** **SWF
7(Volume)**,SWF 8(Trace), SWF
9(Inverse), SWF 10(Transpose), SWF
11(Product), **SWF 12(Det),****
****SWF 13(movie1)****
**

The contents is consists of :

Manual of CD, Contents, Introduction of each Lecture, MPEG file of each introduction, prerequisite, JAVA and Flash, Applications, Exersises, Solutions, All Definitions, All Theorems, Links, BBS,

It is and will be on

http://matrix.skku.ac.kr/sglee/krf/ **for 3 years.
**

<Appedix>

(1) JAVA Source 1, 2

(2) Power Method

Simple Example : Geometric meaning of e.v.

»A=[1 -3;-3 1]

A =

1 -3

-3 1

»eigshow(A)

Next Example : Finding the largest e.v. (Power Method)

Ref : Power Method website

http://math.skku.ac.kr/~sglee/perron_frobenius/perron_frobenius.html

The Power Method Algorithm

» A=[0.98 0.02;0.20 0.80]

A =

0.9800 0.0200

0.2000 0.8000

powplot(A)

(2) Matrix Decomposition in MATHEMATICA

1. LU-Decomposition in MATHEMATICA

2. QR-Factorization in MATHEMATICA

3. SVD Visualization in MATHEMATICA

-9에서 9까지의 Random한 정수를 성분으로 하고 계수(rank)가 5인 8×6 행렬을 만들어 보자.

Singular Value와 Singular Value Decomposition에 의해 구할 수 있는 orthogonal matrix인 U와 V를 구하기 위해 다음 함수를 실행시킨다.

```
←"Tolerance→0"은 가능한 작은
```

singular value를 구하라는 option이다.

```
Out[42]=
```

Out[43]//MatrixForm=

Singular값이 점점 감소함을 볼 수 있다.

Out[45]//MatrixForm=

이는 원래 행렬인 A와 같다. 실제로 빼보면

정확히 0행렬이 나온다.

이상에서 살펴본 바와 같이 MATHEMATICA를 활용하면 SVD이론을 임의로 주어진 행렬에 적용하여 그 분해되는 과정을 단계별로 살펴볼 수 있다. 이는 이론의 전개와 더불어 실제로 분해되는 과정을 software를 이용하여 확인함으로서 학습동기와 숙지도를 높이는데 큰 역할을 한다고 생각한다.