# latar belakang givens matriks

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```					                                      CHAPTER I
INTRODUCTION

1.1 Background
In linear algebra, a rotation matrix is a matrix that is used to perform a
rotation in Euclidean space. To perform a rotation using a rotation matrix R, the
position of each point must be represented by a column vector          , containing the
components of the coordinate of the point. The resulting vector is obtained by using
the matrix multiplication R . Since matrix multiplication has no effect on the zero
vector (i.e., on the coordinates of the origin), rotation matrix can only be used to
describe rotations about the origin of the coordinate system. In general, rotation in
two-dimensional can be written in the form of matrix as follow

.

To find the position of point P(3,2) which is rotated           can be used the matrix
multiplication R , where R is the above matrix and         is [3,2]. Then, we obtain the
new point after it is rotated as follow

.

So the new point is P’(-2,3).
In numerical linear algebra, a Givens rotation is a rotation in a plane spanned
by two coordinate axes. A Givens rotation can be represented in the form of matrix
and called Givens matrix. A Givens matrix is orthogonal which ensures good
theoretical and numerical properties. In practice, Givens matrix is most employed to
sparse matrices, since it affects two rows and two columns only.
Givens matrix plays an important role in scientific computing, especially for
least squares problems, QR decomposition, computational eigen value problems, and
tridiagonalize a given matrix and it is much cheaper (almost half) than the current
tridiagonalization process. Since Givens matrix is very useful while not many people
seminar, especially in terms of its properties and methods to construct it. Hence, we
propose some new matrix which is called generalized Givens matrix. The new matrix
is not orthogonal in general. Necessary and sufficient conditions for orthogonality are
given for generalized Givens matrices.
The organization of this paper is; Givens matrix is introduced, generalized
Givens matrix are established, and condition for orthogonality of generalized Givens
matrix, all of them are established in Chapter III. Based on this background, the
writer is interested in raising a seminar topic entitled “Givens Matrices and Their
Generalizations”.

1.2 Formulated Problems
Based on the above background, we will focus on the following problems:
1. What is Givens matrix?
2. How can we construct generalized Givens matrix?

1.3 Objectives
1. To describe Givens matrix.
2. To invent ways to construct generalized Givens matrix.

1.4 Significances
its generalized form.
2. For writer
Authors can obtain information concerning Givens matrix and its
generalized form.

2

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