An introduction to MATRICES

• Definitions
  
  • Matrix
    
  • Square matrix
    
  • Diagonal matrix
    
  • Row matrix
    
  • Column matrix
    
  • Matrices of the same kind
    
  • The tranpose of a matrix
    
  • 0-matrix
    
  • An identity matrix I
    
  • A scalar matrix S
    
  • The opposite matrix of a matrix
    
  • A symmetric matrix
    
  • A skew-symmetric matrix
    
• The sum of matrices of the same kind
  
  • Sum of matrices
    
  • Sum properties
    
• Scalar multiplication
  
  • Definition
    
  • Properties
    
• Sums in math
  
• Multiplication of a row matrix by a column matrix
  
• Multiplication of two matrices A.B
  
• Properties of multiplication of matrices
  
  • Associativity
    
  • Distributivity
    
  • Theorem 1
    
  • Theorem 2
    
  • Theorem 3
    
  • Theorem 4
    
  • Theorem 5
    

Definitions

Matrix

A matrix is an ordered set of numbers listed rectangular form.

Example. Let A denote the matrix

 
        [2  5  7  8]
        [5  6  8  9]
        [3  9  0  1]

This matrix A has three rows and four columns. We say it is a 3 x 4 matrix.

We denote the element on the second row and fourth column with a_2,4.

Square matrix

If a matrix A has n rows and n columns then we say it's a square matrix.

In a square matrix the elements a_i,i , with i = 1,2,3,... , are called diagonal elements.
Remark. There is no difference between a 1 x 1 matrix and an ordenary number.

Diagonal matrix

 
        [7  0  0]
        [0  5  0]
        [0  0  6]

The matrix is denoted by diag(7 , 5 , 6)

Row matrix

Column matrix

Matrices of the same kind

Matrix A and B are of the same kind if and only if
A has as many rows as B and A has as many columns as B

The tranpose of a matrix

The n x m matrix A' is the transpose of the m x n matrix A if and only if
The ith row of A = the ith column of A' for (i = 1,2,3,..n)
So a_i,j = a_j,i'

 
The transpose of A is denoted T(A) or AT

0-matrix

When all the elements of a matrix A are 0, we call A a 0-matrix.
We write shortly 0 for a 0-matrix.

An identity matrix I

An identity matrix I is a diagonal matrix with all diagonal element = 1.

A scalar matrix S

A scalar matrix S is a diagonal matrix with all diagonal elements alike.
a_1,1 = a_i,i for (i = 1,2,3,..n)

The opposite matrix of a matrix

A symmetric matrix

A square matrix is called symmetric if it is equal to its transpose.
Then a_i,j = a_j,i , for all i and j.

A skew-symmetric matrix

A square matrix is called skew-symmetric if it is equal to the opposite of its transpose.
Then a_i,j = -a_j,i , for all i and j.

The sum of matrices of the same kind

Sum of matrices

To add two matrices of the same kind, we simply add the corresponding elements.

Sum properties

Consider the set S of all n x m matrices (n and m fixed) and A and B are in S.
From the properties of real numbers it's immediate that

• A + B is in S
• the addition of matrices is associative in S
• A + 0 = A = 0 + A
• with each A corresponds an opposite matrix -A
• A + B = B + A

Scalar multiplication

Definition

To multiply a matrix with a real number, we multiply each element with this number.

Properties

Consider the set S of all n x m matrices (n and m fixed). A and B are in S; r and s are real numbers.
It is not difficult to see that:

 
    r(A+B) = rA+rB
    (r+s)A = rA+sA
    (rs)A = r(sA)
    (A + B)T  = AT  + BT
    (rA)T = r. AT

Sums in math

Multiplication of a row matrix by a column matrix

This multiplication is only possible if the row matrix and the column matrix have the same number of elements. The result is a ordinary number ( 1 x 1 matrix).
To multiply the row by the column, one multiplies corresponding elements, then adds the results.
Example.

 
         [1]
[2 1 3]. [2] = [19]
         [5]

Multiplication of two matrices A.B

 
        ci,j = sumk (ai,k.bk,j)

 
[1 2][1 3] = [5 7]
[2 1][2 2]   [4 8]

[1 3][1 2] = [7 5]
[2 2][2 1]   [6 6]

[1 1][2    2] = [0 0]
[1 1][-2  -2]   [0 0]

Properties of multiplication of matrices

Associativity

 
Let D denote B.C, then
        dk,j = sump  bk,p.cp,j        (1)

Let E denote A.D then
        ei,j = sumk ai,k.dk,j (2)

(1) in (2) gives
        ei,j = sumk ai,k.(sump bk,p.cp,j)

<=>     ei,j = sumk,p ai,k.bk,p.cp,j

So the element of the ith row and jth column of A(B.C) is
        sumk,p ai,k.bk,p.cp,j               (3)

 
Let D' denote A.B, then
        di,p' = sumk ai,k.bk,p        (4)

Let E' denote D'C then
        ei,j' = sump di,p'.cp,j       (5)

(4) in (5) gives
        ei,j' = sump (sumk ai,k.bk,p).cp,j

<=>     ei,j' = sumk,p ai,k.bk,p.cp,j

So the element of the ith row and jth column of (A.B)C is
        sumk,p ai,k.bk,p.cp,j               (6)

From (3) and (6)  => A(B.C) = (A.B)C

Distributivity

Theorem 1

Theorem 2

 
        (A.B)T = BT .AT

Theorem 3

 
        A.0 = 0 = 0.A

Theorem 4

Theorem 5

 
if D = diag(a,b,c) then D.D = ( a2 , b2 , c2)
                        D.D.D = ( a3 , b3 , c3)
                        .....