Elementary Matrices

Introduction to Elementary Matrices:

Elementary matrix operation play an significant role in many matrix algebra application, such as finding the inverse of a matrix and solving instantaneous linear equations.

Elementary Operations

The three kind of basic matrix operations.

Interchange two columns.

Multiply each element in a column by a non-zero number.
Multiply a column by a non-zero number and add the result to another column.
While these operation are performed on rows, they are called basic  row operations; and when they are perform on columns, they are called basic column operations.

Types of Elementary Matrices Operations

There are three types of elementary matrices, which match to three types of row operations .

Row switching                RI—>RJ
A row within the matrix can be switched with a new row.

Row multiplication           K RI—>RJ, where k?0            
Each element in a row can be multiplied by a non-zero even.

Row addition              RI+ KRj—> RI
A line can be replaced by the sum of the  row and a multiple of a new row.

The elementary matrix for any row operation is obtained by execute the operation on an identity matrix.

Elementary Operation Notation

In many reference, you resolve encounter a compacted notation to describe elementary operations. That notation is given by

Row operations

1. Interchange rows i and j Ri <--> Rj

2. Multiply row i by s, where s ? 0 sRi --> Ri

3. Add s times row i to row j sRi + Rj --> Rj

Column operations

1. Interchange columns i and j Ci <--> Cj

2. Multiply column i by s, where s ? 0 sCi --> Ci

3. Add s times column i to column j sCi + Cj --> Cj

Rules on Elementary Matrices

Elementary Matrices

Now, imagine R has 1 ,n>n elementary square matrix obtained from the identity matrix by performing arts an basic operation. A result, we have three types of elementary matrices, every subsequent to a type of elementary operations:

Transposition matrix Tij : an matrix obtain from In with rows i and j switch,

Basic diagonal matrix Di(r) : a diagonal matrix whose entries are 1 except in cell (i' > i)  , whose entry is a non-zero element r of R

Row substitution matrix Eij(s) : In+sUij , where s > R and Uij is a matrix unit with i=j  .



Examples based on elementary matrices:

Ex 1: Construct a 3 × 2 matrix whose entries are given by aij = i - 2j

Sol : The general 3 x 2 matrix is of the form,

A= [aij]  =`[[a_11,a_12],[a_21,a_22],[a_31,a_32]]`

where ,i= 1,2,3(rows),j= 1,2 (columns)

it is given that aij = i-2j

a11 =1-2 = -1

a12 =1-4 = -3

a21  =2-2 =0

a22 =2-4 = -2

a31 = 3-2 =1

a32 = 3-4 = -1

Therefore , The required matrix  is  A = `[[-1,-3],[0,-2],[1,-1]]`

Ex 2: Elementary column operation on the term “A”, where the identity is used in matrix   `A= [[1,6], [4,1], [0,1]]` and identity matrix `x = [[1,0], [0 ,1]]`

Sol:

Given: Identity matrix `x = [[1,0], [0 ,1]]`

`A= [[1,6], [4,1], [0,1]]`

Step 1:  The given identity matrix x is interchanged of first and second column and the finding result is named it   (x1).

Step 2: To do the elementary column operation on given `A= [[1,6], [4,1], [0,1]]`

Step 3: The operation in matrix is done by [row *column]

`X= [[1,0], [0,1]]`

The first and second column is interchanged and the result is,

`X_1= [[0,1], [1,0]].`

Now, the found X1 is calculated with given “A”

`A= [[1,6], [4,1], [0,1]]`

`X_1= [[0,1], [1,0]]`

`A*X_1` for calculation elementary column operation on

`A*X_1`

` [[1,6], [4,1], [0,1]] * [[0,1], [1,0]]`

` [[1*0+6*1 , 1*1+6*0],[4*0+1*1 , 4*1+1*0],[0*0+1*1, 0*1+1*0]]` 

`[[0+6, 1+0], [0+1,4+1], [0+1, 0+0]]`

`[[6,1],[1,4],[1,0]]`

Hence ,the elementary column operation for the given value `A=[[6,1],[1,4],[1,0]]`