Theory and application of Eignvalue and Eigenvector
CHAPTER ONE
INTRODUCTION
1.1 BACKGROUND OF STUDY:
According to concise encyclopaedia of science and technology (2004); Mathematics is not a branch of science, it’s the language of science in a deep sense, an indispensable medium by which and within which science expresses, formulates, continues and communicates itself. Mathematics is frequently encountered in association and interaction with astronomy, physics and other branches of natural science and it also has deep rooted affinities to the humanities.
“Mathematics is therefore defined as a realm of knowledge entirely unto itself and one of considerable scope”.
As the earth rotates every arrow pointing out from the centre also rotates, except those arrows that lie on the axis of rotation.
Consider the transformation of the earth after one hour of the rotation, an arrow from the centre of the of the earth to the geographic south pole would be an eigenvector of this transformation, but an arrow from the centre of the earth to anywhere on the equator would not be an eigenvector.
Since the arrow pointing at the pole is not stretched by the rotation of the earth, its eigenvalue is 1 (one). Eigenvalues are introduced in the context of linear algebra or matrix theory. The introduction and development of the notation of matrix and system of linear algebra followed the development of determinant, which rose from the study of coefficient system of linear equations.
Leibnitz, one of the founders of calculus used determinant in 1693 an Cramer presented his determinant - based formula system of solving linear equations (known as Cramer’s’ rule) in 1750.
In 1848, James J Sylvester introduced the term “matrix” which was Latin word for Womb, as a name of an array of numbers.
Historically, however eigenvalue and eigenvectors arose in the study of quadratic forms and differential equations. Euler studied the rotational motion of a rigid body and discovered the importance of principal axes. Lagrange realized that
INTRODUCTION
1.1 BACKGROUND OF STUDY:
According to concise encyclopaedia of science and technology (2004); Mathematics is not a branch of science, it’s the language of science in a deep sense, an indispensable medium by which and within which science expresses, formulates, continues and communicates itself. Mathematics is frequently encountered in association and interaction with astronomy, physics and other branches of natural science and it also has deep rooted affinities to the humanities.
“Mathematics is therefore defined as a realm of knowledge entirely unto itself and one of considerable scope”.
As the earth rotates every arrow pointing out from the centre also rotates, except those arrows that lie on the axis of rotation.
Consider the transformation of the earth after one hour of the rotation, an arrow from the centre of the of the earth to the geographic south pole would be an eigenvector of this transformation, but an arrow from the centre of the earth to anywhere on the equator would not be an eigenvector.
Since the arrow pointing at the pole is not stretched by the rotation of the earth, its eigenvalue is 1 (one). Eigenvalues are introduced in the context of linear algebra or matrix theory. The introduction and development of the notation of matrix and system of linear algebra followed the development of determinant, which rose from the study of coefficient system of linear equations.
Leibnitz, one of the founders of calculus used determinant in 1693 an Cramer presented his determinant - based formula system of solving linear equations (known as Cramer’s’ rule) in 1750.
In 1848, James J Sylvester introduced the term “matrix” which was Latin word for Womb, as a name of an array of numbers.
Historically, however eigenvalue and eigenvectors arose in the study of quadratic forms and differential equations. Euler studied the rotational motion of a rigid body and discovered the importance of principal axes. Lagrange realized that
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