MATHEMATICS (T) COURSEWORK 954/4 PAPER 4

TITLE: “Usage of Matrix”

NAME : Chuah Rui Han

I/C : 950625-14-6077

SCHOOL : SMK Aminuddin Baki

TEACHER : Mr Murugeson K.G.

Definition

Matrix is a set of numbers arranged in rows and columns in a rectangular array and enclosed by a pair of brackets.

"Matrix" is the Latin word for womb. It can also mean more generally any place in which something is formed or produced.

History

The orgins of mathematical matrices lie with the study of systems of simultaneous linear equations. An important Chinese text from between 300 BC and AD 200, Nine Chapters of the Mathematical Art (Chiu Chang Suan Shu), gives the first known example of the use of matrix methods to solve simultaneous equations, including the concept of determinants, over 1000 years before its publication by the Japanese mathematician Seki in 1683[90] and the German mathematician Gottfried Leibniz in 1693.

More uses of matrix-like arrangements of numbers appear in chapter eight of the text, "Methods of rectangular arrays," 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 Carl Friedrich Gauss (1777-1855), also known as Gaussian elimination.

The term "matrix" for such arrangements was introduced in 1850 by James Joseph Sylvester.

Early matrix theory emphasized determinants more strongly than matrices and an independent matrix concept akin to the modern notion emerged only in 1858, with Cayley's Memoir on the theory of matrices.

Today, matrices are used not simply for solving systems of simultaneous linear equations, but also for describing the quantum mechanics of atomic structure, designing computer game graphics, analyzing relationships, and even plotting complicated dance steps!

Row echelon form

In linear algebra, a matrix is in echelon form if it has the shape resulting of a Gaussian elimination. Row echelon form means that Gaussian elimination has operated on the rows and column echelon form means that Gaussian elimination has operated on the columns. In other words, a matrix is in column echelon form if its transpose is in row echelon form. Therefore only row echelon forms are considered in the remainder of this article. The similar properties of column echelon form are easily deduced by transposing all the matrices.

Specifically, a matrix is in row echelon form if

All nonzero rows (rows with at least one nonzero element) are above any rows of all zeroes (all zero rows, if any, belong at the bottom of the matrix).

The leading coefficient (the first nonzero number from the left, also called the pivot) of a nonzero row is always strictly to the right of the leading coefficient of the row above it.

All entries in a column below a leading entry are zeroes (implied by the first two criteria).[1]

Some texts add the condition that the leading coefficient of any nonzero row must be 1.[2]

This is an example of a 3×5 matrix in row echelon form:

Reduced row echelon form

A matrix is in reduced row echelon form (also called row canonical form) if it satisfies the following conditions:

It is in row echelon form. Every leading coefficient is 1 and is the only nonzero entry in its column.[3]

The reduced row echelon form of a matrix may be computed by Gauss–Jordan elimination. Unlike the row echelon form, the reduced row echelon form of a matrix is unique and does not depend on the algorithm used to compute it.

This is an example of a matrix in reduced row echelon form:

Note that this does not always mean that the left of the matrix will be an identity matrix. For example, the following matrix is also in reduced row-echelon form:

For matrices with integer coefficients, the Hermite normal form is a row echelon form that may be computed using Euclidean division and without introducing any rational number nor denominator. On the other hand, the reduced echelon form of a matrix with integer coefficients generally contains non-integer entries.

Transformation to row echelon form

By means of a finite sequence of elementary row operations, called Gaussian elimination, any matrix can be transformed to row echelon form. Since elementary row operations preserve the row space of the matrix, the row space of the row echelon form is the same as that of the original matrix.

The resulting echelon form is not unique; for example, any multiple by a scalar of a matrix in echelon form is also an echelon form of the same matrix. However, every matrix has a unique reduced row echelon form. This means that the nonzero rows of the reduced row echelon form are the unique reduced row echelon generating set for the row space of the original matrix.

Systems of linear equations

A system of linear equations is said to be in row echelon form if its augmented matrix is in row echelon form. Similarly, a system of equations is said to be in reduced row echelon form or canonical form if its augmented matrix is in reduced row echelon form.

The canonical form may be viewed as an explicit solution of the linear system. In fact, the system is inconsistent, if and only if one of the equations of the canonical form is reduced to 1 = 0. Otherwise, regrouping in the right hand side all the terms of the equations, but the leading ones expresses the variables corresponding to the pivots as constants or linear functions of the other variables, if any.

Row equivalence

In linear algebra, two matrices are row equivalent if one can be changed to the other by a sequence of elementary row operations. Alternatively, two m × n matrices are row equivalent if and only if they have the same row space. The concept is most commonly applied to matrices that represent systems of linear equations, in which case two matrices of the same size are row equivalent if and only if the corresponding homogeneous systems have the same set of solutions, or equivalently the matrices have the same null space.

Because elementary row operations are reversible, row equivalence is an equivalence relation. It is commonly denoted by a tilde (~).

There is a similar notion of column equivalence, defined by elementary column operations; two matrices are column equivalent if and only if their transpose matrices are row equivalent. Two rectangular matrices that can be converted into one another allowing both elementary row and column operations are called simply equivalent.

Elementary row operations

An elementary row operation is any one of the following moves:

1. Swap: Swap two rows of a matrix.2. Scale: Multiply a row of a matrix by a nonzero constant.3. Pivot: Add a multiple of one row of a matrix to another row.

Two matrices A and B are row equivalent if it is possible to transform A into B by a sequence of elementary row operations.

Row space

The row space of a matrix is the set of all possible linear combinations of its row vectors. If the rows of the matrix represent a system of linear equations, then the row space consists of all linear equations that can be deduced algebraically from those in the system. Two m × n matrices are row equivalent if and only if they have the same row space.

For example, the matrices

are row equivalent, the row space being all vectors of the form . The corresponding systems of homogeneous equations convey the same information:

In particular, both of these systems imply every equation of the form

Equivalence of the definitions

The fact that two matrices are row equivalent if and only if they have the same row space is an important theorem in linear algebra. The proof is based on the following observations:

1. Elementary row operations do not affect the row space of a matrix. In particular, any two row equivalent matrices have the same row space.

2. Any matrix can be reduced by elementary row operations to a matrix in reduced row echelon form.

3. Two matrices in reduced row echelon form have the same row space if and only if they are equal.

This line of reasoning also proves that every matrix is row equivalent to a unique matrix with reduced row echelon form.

Additional properties

Because the null space of a matrix is the orthogonal complement of the row space, two matrices are row equivalent if and only if they have the same null space.

The rank of a matrix is equal to the dimension of the row space, so row equivalent matrices must have the same rank. This is equal to the number of pivots in the reduced row echelon form.

A matrix is invertible if and only if it is row equivalent to the identity matrix.

DECLARATION:

This is to certify that this assignment report submitted is based on my own work except for those which have been referred and acknowledged duly in this report.

Signature : .............................

Name : Chuah Rui Han

I/C Number : 950625-14-6077

Date : 30-9-2013