Matrices And It’s Application To Science And Technology

Project and Seminar Material for Mathematics and Statistics

Abstract

This study was carried out to assess matrices and its application to science and technology. Scientific and other non-patent references (NPRs) in patents are important tools to analyze interactions between science and technology. This study organizes a empirical literature review with 514,894 USPTO patents granted globally in 1974, 1982, 1990, 1998 and 2016. There are 165,762 patents with at least one reference to science and engineering (S&E) literature, from a total of 1,375,503 references.

Through a lexical analysis, 71.1% of this S&E literature is classified by S&E fields. These data serve as the basis for the elaboration of global and national 3-dimensional matrices (technological domains, S&E fields and number of references). Three indicators are proposed to analyze these matrices, allowing us to identify patterns of structured growth that differentiate developed and non-developed countries.

This differentiation informs suggestions for public policies for development, emphasizing the need for an articulation between the industrial and technological dimension and scientific side. The intertwinement of these two dimensions is a key component of developmental policies for the twenty-first century.

Table Of Content

Preliminary Page(s)

Title page
Certification page
Dedication
Acknowledgement
Abstract
Table of content

Chapter One

1.0 Introduction

1.1 Background Of The Study
1.2 Statement Of Problem
1.3 Aims And Objectives

Chapter Two

2.0 Literature Review

2.1 Conceptual Review
2.2 The Matrix Of Science And Technology Interactions
2.3 Previous Literature On These Matrices

Chapter Three

3.0 Research Methodology

3.1 Database Preparation And The Three Indicators

Chapter Four

4.0 Results And Discussion

Chapter Five

5.0 Conclusion And Recommendation

5.1 Conclusion
5.2 Recommendations
References

Chapter One

General Introduction

1.1 Background Of The Study

In order to unfold the history of Matrices and Its Applications, the influence of matrices in the mathematical world is spread wide because it provides an important base to many of the principles and practices. It is important that we first determine what matrices is. As such, this definition is not a complete and comprehensive answer, but rather a broad definition loosely wrapping itself around the subject.

“Matrix” is the Latin word for womb, and it retains that sense in English. It can also mean more generally any place in which something is formed or produced.

The origin of mathematical matrices lies with the study of systems of simultaneous linear equations. An important Chinese text from between 300Bc and Ad 200, nine chapters of the mathematical art, gives the first known example of the use of matrix methods to solve simultaneous equations.(Laura Smoller, 2012)

In the treatises seventh chapter “too much and not enough”, the concept of a determinant first appears, nearly two milkman before its supposed inventions by the Japanese mathematician SEKI KOWA in 1683 or his german contemporary GOTTFRIED LEIBNIZ (who is also credited with the invention of differential calculus, separately from but simultaneously with Isaac Newton).

More uses of matrix-like arrangements of numbers appears in which a method is given for solving simultaneous equations using a counting board that is mathematically identical to the modern matrix method of solution outlined by Carh Fridrich Gauss (1777-1855) also known as Gaussan Elimination.(Vitull marie 2012 )

This project seeks to give an overview of the history of matrices and its practical applications touching on the various topics used in concordance with it.

Around 4000 years ago, the people of Babylon knew how to solve a simple2X2 system of linear equations with two unknowns. Around 200 BC, theChinese published that “Nine Chapters of the MathematicalArt,” they displayed the ability to solve a 3X3 system of equations (Perotti). The power and progress in Matrices and its application did not come to fruition until the late 17th century.

The emergence of the subject came from determinants, values connected to a square matrix, studied by the founder of calculus, Leibnitz, in the late 17th century. Lagrange came out with his work regarding Lagrange multipliers, a way to “characterize the maxima and minima multivariate functions.” (Darkwing) More than fifty years later, Cramer presented his ideas of solving systems of linear equations based on determinants more than50 years after Leibnitz (Darkwing). Interestingly enough, Cramer provided no proof for solving an n x n system.

As mentioned before,Gauss work dealt much with solving linear equations themselves initially, but did not have as much to do with matrices. In order for matrix algebra to develop, a proper notation-or method of describing the process was necessary.Also vital to this process was a definition of matrix multiplication and the facets involving it.“The introduction of matrix notation and the invention of the word matrix were motivated by attempts to develop the right algebraic language for studying determinants. In 1848, J.J. Sylvester introduced the term “matrix,” the Latin word for womb, as a name for an array of numbers. He used womb, because see, linear algebra has become more relevant since the emergence of calculus even though its foundational equation of ax+ b=0 dates back centuries.

Euler brought to light the idea that a system of equations doesn’t necessarily have to have a solution. He recognized the need for conditions to be placed upon unknown variables in order to find a solution. The initial work up until this period mainly dealt with the concept of unique solutions and square matrices where the number of equations matched the number of unknowns.

With the turn into the 19th century Gauss introduced a procedure to be used for solving a system of linear equations. His work dealt mainly with the linear equations and had yet to bring in the idea of matrices or their notations. His efforts dealt with equations of differing numbers and variables as well as the traditional pre-19th century works of Euler, Leibnitz, andCramer. Gauss’ work is now summed up in the term Gaussian elimination. This method uses the concepts of combining, swapping, or multiplying rows with each other in order to eliminate variables from certain equations. After variables are determined, the student is then to use back substitution to help find the remaining unknown variables.

Reviewed a matrix as a generator of determinants(Tucker, 1993).The other part, matrix multiplication or matrix algebra came from the work of Arthur Cayley in 1855.

Cayley’s defined matrix multiplication as, “the matrix of coefficients for the composite transformation T2T1 is the product of the matrix for T2times the matrix of T1”(Tucker, 1993). His work dealing with Matrix multiplication culminated in his theorem, the Cayley-Hamilton Theorem. Simply stated, a square matrix satisfies Matrices at the end of the 19th century were heavily connected with Physics issues and for mathematicians, more attention was given to vectors as they proved to be basic mathematical elements. With the advancement of technology using the methods of Cayley, Gauss, Leibnitz, Euler, and others determinants and linear algebra moved forward more quickly and more effective. Regardless of the technology though Gaussian elimination still proves to be the best way known to solve a system of linear equations (Tucker,1993).

The influence of matrices and it’s applications in the mathematical world is spread wide because it provides an important base to many of the principles and practices. Some of the things Matrices is used for are to solve systems of linear format, to find least-square best fit lines to predict future outcomes or find trends, to encode and decode messages. Other more broad topics that it is used for are to solve questions of energy in Quantum mechanics. It is also used to create simple everyday household games likeSudoku. It is because of these practical applications that Matrices has spread so far and advanced. The key, however, is to understand that the history of linear algebra provides the basis for these applications.

Although linear algebra is a fairly new subject when compared to other mathematical practices, its uses are widespread. With the efforts of calculus-savvy Leibnitz the concept of using systems of linear equations to solve unknowns was formalized. Other efforts fromScholars like Cayley. Euler, Sylvester, and others changed matrices into the use of linear algebra to represent them. Gauss brought his theory to solve systems of equations proving to be the most effective basis for solving unknowns.

Technology continues to push the use further and further, but the history of matrices and its application continues to provide the foundation. Even though every few years companies update their textbooks, the fundamentals stay the same.(laura smoller (2001)[9].

1.2 Statement Of Problem

Due to the great need of security for passing sensitive information from one person to another or from one organization to another through electronic technology, there is need for cryptography as a solution to this problem.

Also in economics this research work is going to discuss how Leontief model is used to represent the economy as a system of linear equation so as to calculate the gross domestic products and goods production efficiently.

1.3 Aims And Objectives

To apply matrices to Cryptography, Economic Models and system of Linear Equations
To improve the methods at which increase in production out-put can be achieved
To show ways at which sensitive information can be passed across mathematically.
To disseminate this improved methods to the relevant communities and end use

Chapter Five

5.0 Conclusion And Recommendation

5.1 Conclusion

The investigation of S&E literature citations in patents is a useful tool to investigate the nature of science and technology linkages both in developed and under-developed countries, inter alia for it allows dialogues with other tools available both for developed (Cohen et al. 2002) and under-developed countries (Rapini 2007).

The scientific content of technology, as measured by S&E literature citations in patents, is increasing steady both in developed and under-developed countries, but the nature of these increases differs across countries and levels of development.

The elaboration of three dimensional matrices (OST-technological domain, ISI-disciplines, and number of references per matrix cell) for each country and each year is a powerful tool for evaluation of the stage and the dynamics of interactions between science and technology.

The indicators about Matrix Filling and Matrix Rugosity provide important qualitative insights about these interactions. Once these qualitative insights are available, the implications for development are not difficult to see.

The problem is not only the scarcity of patents, but also the quality of those important but few patents from developing countries, the countries within regimes I and II, in our previous work (Ribeiro et al. 2016a, b).

The inter-temporal correlations between matrices’ surfaces are the basis for the identification of patterns of structured growth, a key difference between countries within regime III (mature NSIs) and the rest (immature NSIs).

5.2 Recommendations

Given these conclusions, there are important implications for development, in an era when science, technology and their linkages matter:

The interconnections between science and technology may indicate which S&E fields should be supported for specific industrial policies, and provide policy makers a tool for designing industrial policies that take into account the interactions between science and technology as a key factor for development.
The role of persistence over time must be stressed. This involves long-term planning by firms and public agencies (probably interacting with policies to mitigate the high mortality rates of new firms, firms so necessary to change the technological landscape of underdeveloped countries).
More evidence in favor of a very simple argument: a broad science and technology infrastructure is necessary for development, and this necessity grows over time. The argument is very simple: to catch up, a country needs to improve its innovation capabilities. Over time, the scientific content of technology is increasing. Therefore, inter alia, a greater and deeper scientific infrastructure is necessary to support these innovative activities. This process seems to be unavoidable and demands larger investments in science in LDCs than have been done so far.
New arguments for the necessary combination between industrial policies and science and technology policies: the evidence presented in this paper suggests that for a quantitative increase in patent figures, a precondition is a correspondent growth in science and engineering publications. No quantitative increase in patent figures is possible without a qualitative improvement in the patents generated; in other words, in their science and engineering content. Therefore, this paper suggests that dynamically, over time, there is a deep relationship between the quantity of patents and the quality of these patents.