Matrix
A matrix is an ordered set of numbers listed rectangular form.
Example. Let A denote the matrix
[2
[5
[3
5
6
9
7
8
0
8]
9]
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 a2,4.
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 ai,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
A diagonal matrix is a square matrix with all de non-diagonal elements 0.
The diagonal matrix is completely denoted by the diagonal elements.
Example.
[7
[0
[0
0
5
0
0]
0]
6]
The matrix is denoted by diag(7 , 5 , 6)
Row matrix
A matrix with one row is called a row matrix
Column matrix
A matrix with one column is called a 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 ai,j = aj,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.
a1,1 = ai,i for (i = 1,2,3,..n)
The opposite matrix of a matrix
If we change the sign of all the elements of a matrix A, we have the
opposite matrix -A.
If A' is the opposite of A then ai,j' = -ai,j, for all i and j.
A symmetric matrix
A square matrix is called symmetric if it is equal to its transpose.
Then ai,j = aj,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 ai,j = -aj,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
Because in the following, there is an intensive use of the properties of
sums, the reader who is not familiar with these properties must read first
Sums in math .
Remark. In this html document, for convenience, we'll write the word
sum instead of the sigma sign.
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
This product is defined only if A is a (l x m) matrix and B is a (m x n)
matrix.
So the number of columns of A has to be equal to the number of rows of
B.
The product C = A.B then is a (l x n) matrix.
The element of the ith row and the jth column of the product is found by
multiplying the ith row of A by the jth column of B.
ci,j = sumk (ai,k.bk,j)
Example.
[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
[1 1][-2
2] = [0 0]
-2]
[0 0]
From these examples we see that the product is not commutative and that
there are zero divisors.
Properties of multiplication of matrices
Associativity
If the multiplication is defined then A(B.C) = (A.B)C holds for all
matrices A,B and C.
Proof:
We'll show that an element of A(B.C) is equal to the corresponding
element of (A.B)C
First we calculate the element of the ith row and jth column of A(B.C)
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)
Now we calculate the element of the ith row and jth column of (A.B)C
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)
Distributivity
=> A(B.C) = (A.B)C
If the multiplication is defined then A(B+C) = A.B+A.C and (A+B).C =
A.C+B.C holds for all matrices A,B and C. This theorem can be proved
in the same way as above.
Theorem 1
For each A, there is always an identity matrix E and an identity matrix E'
so that A.E = A and E'.A = A If A is a square matrix, E = E'.
Theorem 2
(A.B)T = BT .AT
This theorem can be proved in the same way as above.
Theorem 3
If the multiplication is defined then for each A
A.0 = 0 = 0.A
Theorem 4
r and s are real numbers and A , B matrices. If the multiplication is
defined then (rA)(sB) = (rs)(AB) This theorem can be proved in the same
way as above.
Theorem 5
if D = diag(a,b,c) then D.D = ( a2 , b2 , c2)
D.D.D = ( a3 , b3 , c3)
.....
This property can be generalised for D = diag(a,b,c,d,e,...,l).