Csci 1802 Computer Systems
Matrices Lecture 1
Learning outcomes
Be able to add and multiply matrices.
Know some uses of matrices in a
computing context
Matrices
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Csci 1802 Computer Systems
Introduction
Vectors and matrices are
 data structures
 for collections of data all of the same type
Examples of uses of vectors and matrices:
1. Vector of names
2. Matrix of marks
 Adams 


 Brown 
 Chauhan 


 Davis 
 Edwards 


 65

 70
 66

 47
 58

3.
a
c
Digraph
and its
72
75
52
55
60
54
65
57
45
47
66 

64 
56 

35 
52 
Matrix representation
b
0

0
1

0

1 0 1

0 0 0
0 0 0

1 0 0 
d
In programming
 vectors are called one-dimensional arrays
 matrices are called two-dimensional arrays
 can also go into 3 or more dimensions
In this maths module, we’re not going higher than two dimensions.
Parallel processors.
Computers designed to do certain types of calculations quicker by working on more than
one pair of values at a time.
e.g. Vector Processors and Array Processors
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Csci 1802 Computer Systems
In the sections that follow, we shall be looking at
- Vector and Matrix notation
- Arithmetic of Vectors and Matrices
- Application to nets
- Application to computer graphics
- Application to codes
(Note: matrix is the singular of matrices i.e. you talk about one matrix or several
matrices)
Vectors
Row Vector
[1 4 8 2 4]
Column vector
 3
 
5
1
 
 2
 
To give a vector a name, use underline to indicate that it is a vector.
e.g.
x=[1243]
Vector Arithmetic
Addition and Subtraction
You can only do addition and subtraction between vectors that are of the same type and
with the same number of elements.
Examples:
1)
[1 3 2 4] + [3 -1 2 6] =
2)
 2   6 8
     
 3    1   4
  4  4  0
     
[4 2 4 10]
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Csci 1802 Computer Systems
3)
If
1
 
  1
1
 
x=
and y =
 8 
 
 3 
  2
 
 
 
then
x+y=
9
 
2
 1
 
and
x-y=
 7
 
  4
 3 
 
Exercises:
1.
x = [1 1 4 2] and y = [1 -1 5 4] and z = [3 3 0 1]
Calculate (i)
2.
y=
x+y
(ii)
y-z
(iii)
x+y+z
 1
 
  2
  3
 
and
z=
 2
 
 3
 3
 
-
Calculate (i)
(ii)
y-z
-y - z
(iii) z + y
Vector Notation
Labelling elements in vectors:
Row vector x
[x1 x2 x3 x4 ]
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Csci 1802 Computer Systems
 x1 
 
 x2 
x 
 3
Column vector x
e.g. If x and y are both row vectors having 5 elements
then x + y = [x1+y1 x2+y2 x3+y3 x4+y4 x5+y5]
The result of
z=x+y
is
zi = xi + yi
for i = 1,2, .. n where n
is the size of the vector
Matrices
Matrix Notation
Matrices have both rows & columns, so we use 2 subscripts
 first subscript is the row label
 second subscript is the column label.
e.g. If matrix M is a 3 x 5 matrix (i.e. 3 rows by 5 columns)
1
2
3
4
5
row labels
1
column labels
m11
m12
m13
m14
m15
2
3
m21
m31
m22
m32
m23
m33
m24
m34
m25
m35
We use
 capital letters for the matrix name
 can also use e.g [aij] to denote matrix A.
i.e. A = [aij]
A Square Matrix is a matrix which has the same number of rows and columns.
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Csci 1802 Computer Systems
A Diagonal Matrix is a square matrix which has non-zero element values only on the
leading diagonal of the matrix and all the other elements are zero.
e.g.
3 0 0


0 1 0
0 0 2


A Symmetric Matrix is one where the non-diagonal elements have a 'mirror-image'
across the leading diagonal
5 2 1


e.g.  2 8 0 
1 0 7


The Transpose of a matrix is the matrix formed by interchanging the rows and the
columns. The transpose of A is denoted by A' (or sometimes AT)
1 2


then A' =  0 3 
5 6


 1 0 5

A = 
 2 3 6
e.g. If
12.5.2 Exercises
1.
State whether each of the following matrices is
(a) square, (b) diagonal, (c) symmetric
(i)
1 5 7


0 4 0
0 5 6


2.
If matrix A = [aij] and is equal to matrix (i) above, what are the values of
(a)
a22 =
 1 4 2


(ii)  4 4 6 
 2 6 0


(b)
1 3


(iii)  3 1 
0 4


a13 =
13 0 

(iv) 
 0 1
(c)
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a23 =
Csci 1802 Computer Systems
3.
What is the transpose of A?
A
2 4 6 8


=  1 1 2 4
 5 3 1 3


A' =
Matrix Arithmetic
Addition and Subtraction


can only add or subtract matrices of the same size & shape
simply add or subtract the corresponding elements to get a matrix of the same size
and shape as the original ones
e.g.
 1 4   2 7   3 11 

 
 

 2 2  +  1 4 =  1 6 
 5  5   9 3  14  2 

 
 

Thus, the result of
C = A + B ( [cij] = [aij] + [bij] )
is
cij
aij + bij
&
for i = 1,2, .. n
for j = 1,2, .. m
(
i,j : 1..n · cij = aij + bij
=
where n is the number of rows
where m is the number of cols
)
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Csci 1802 Computer Systems
Matrix Multiplication
 only certain sizes and shapes of vectors and matrices can be multiplied together
 the order in which the matrices or vectors are multiplied is significant
 the dimensions of the resultant matrix may be different from the original ones
 the number of columns in the first matrix or vector must be the same as the number
of rows in the second matrix or vector
Example:
To multiply
x.M
where
x is a row vector of 3 elements and
M is a 3 x 2 matrix
 1 2


2 4 1 0 1   4 11
 2 3


The result is a row vector of 2 elements.
What we did to obtain each element in the result r was
To get r11 take the 1st row of the vector and the 1st column of the matrix, then
r11
=
x11 . m11 +
x12 . m21 +
x13 . m31
To get r12 take the 1st row of the vector and the 2nd column of the matrix, then
r12
=
x11 . m12 +
x12 . m22 +
x13 . m32
Remember, [m x n] means a matrix with m rows and n columns
And for multiplication we go
 Along the rows of the first matrix
 Down the columns of the second matrix
If the number of columns in the first matrix or vector ISN’T the same as the number of
rows in the second matrix or vector they cannot be multiplied together (i.e. they are not
compatible)
The size of the answer can be predicted: [m x n] [n x p] gives answer [m x p]
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Csci 1802 Computer Systems
Example
 1 2
  1  1  2  2  3  2 1  2  2  1  3  2  11 10 
 1 2 3 

 2 1   
  

 2 1 2  2 2   2  1  1  2  2  2 2  2  1  1  2  2   8 9 


This layout may help:
1st matrix
1
2
2
1
3
2
11
8
10
9
x
1
2
2
2
1
2
Result
2nd matrix
This emphasizes that to get the result for (e.g.) element R21 you use row 2 and column 1
Exercises
1. Where possible do the following multiplications. State which ones are not compatible.
(a)
 1 1  2 3 1 



 2 3  1 2 2 
(b)
5
(c)
 1 0  5 4 1 



0
2
2
0
3



2 3 1

2 2
 1 2 2
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Csci 1802 Computer Systems
2. Multiply the following matrices A.B. Show that B.A gives a different result.
1 2 2


A =  0 1 3
3 1 1


 3 0 0


B =  2 1 4
0 3 1


1 2 2  3 0 0



A.B =  0 1 3   2 1 4  =
3 1 1 0 3 1



B.A =
 3 0 0 1 2 2



 2 1 4  0 1 3 =
0 3 1 3 1 1



Inverse of a Matrix
There is a special case for square matrices where
A.B = B.A.
This is when matrix B is the inverse of matrix A.
(It also follows that matrix A is the inverse of matrix B.)
The inverse of matrix A is denoted by A-1.
A-1 =
1
A
The result of the multiplications A.A-1 and A-1.A is the identity matrix I. This is a
special type of diagonal matrix where the diagonal elements have the value 1.
A . A-1 = A-1. A
=
I
=
 1 0 .. 0 


 0 1 .. 0 
. . . .


 0 .. 0 1 


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Csci 1802 Computer Systems
Note that not all square matrices have an inverse.
12.7.1 Exercise
Prove that the following matrices are the inverse of each other by multiplying A.B and
B.A and checking that the result in both cases is the identity matrix.
A
1 3 3 


= 1 4 3 
1 3 4 


B
=
 7  3  3


0 
 1 1
 1 0
1 

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