GG 313 Lecture 11
Chapter 3
Linear (Matrix) Algebra
Sept 27, 2005
Homework due today:
1st problem: “…do these data support the claim
that on average higher concentrations were
obtained before cleaning versus after cleaning?”
What is the hypothesis? Null hypothesis?
One-sided or 2-sided? What is your
conclusion?
2nd problem:”…test whether the number of days
with ozone standards exceeded is different for urban
versus rural sites.” What is the hypothesis? Null
hypothesis? One-sided or 2-sided? What is
your conclusion?
Degrees of freedom:
The degrees of freedom, , are the number of
values in the final calculation of a statistic that are
allowed to vary.
MEAN: (x1+….+xn)/n
n values can vary, so =n
VARIANCE (std.dev.): sum((xi-mean)2)/(n)
How many free variables? If we know all of the xi
values except one, we can calculate the last one.
Multiply the mean2 by n and subtract each known
xi2. The remainder will be the unknown xi2. Since
there are n-1 values free to vary, =n-1 for the
variance.
Degrees of freedom: When we get to curve fitting,
we will have two statistics - slope and y-intercept
for example, which will reduce the degrees of
freedom by 2.
Example: It takes two points to define a straight
line with no degrees of freedom. A circle requires
three points to be defined with no degrees of
freedom. This means that if you ONLY have two
points on a curve, it might have any shape, but you
can define a straight line that will fit those points
exactly. With three points, you can define a circle
exactly.
MATRIX ALGEBRA
Some history*:
The beginnings of matrices and determinants goes
back to the second century BC although traces
can be seen back to the fourth century BC.
However it was not until near the end of the 17th
Century that the ideas reappeared and
development really got underway.
*http://wwwgroups.dcs.stand.ac.uk/~history/HistTopics/Matrices_and_determinants.html
It is not surprising that the beginnings of matrices and
determinants should arise through the study of systems
of linear equations. The Babylonians studied problems
which lead to simultaneous linear equations and some
of these are preserved in clay tablets which survive.
For example a tablet dating from around 300 BC
contains the following problem:
-There are two fields whose total area is 1800 square
yards. One produces grain at the rate of 2/3 of a bushel
per square yard while the other produces grain at the
rate of 1/2 a bushel per square yard. If the total yield is
1100 bushels, what is the size of each field.
The Chinese, between 200 BC and 100 BC, came much closer to
matrices than the Babylonians. Indeed it is fair to say that the text
Nine Chapters on the Mathematical Art written during the Han
Dynasty gives the first known example of matrix methods. First a
problem is set up which is similar to the Babylonian example given
above:
-There are three types of corn, of which three bundles
of the first, two of the second, and one of the third make
39 measures. Two of the first, three of the second and
one of the third make 34 measures. And one of the first,
two of the second and three of the third make 26
measures. How many measures of corn are contained of
one bundle of each type?
Now the author does something quite remarkable.
He sets up the coefficients of the system of three
linear equations in three unknowns as a table on a
'counting board'.
bundles
1
2
3 type 1 corn
2
3
2 type 2
3
1
1 type 3
26 34 39 measures
the author, writing in 200 BC, instructs the reader
to multiply the middle column by 3 and subtract
the right column as many times as possible, the
same is then done subtracting the right column
as many times as possible from 3 times the first
column. This gives
0
0
3
4
5
2
8
1
1
39
24
39
Next the left most column is multiplied by 5
and then the middle column is subtracted as
many times as possible. This gives
0
0
3
0
5
2
36
1
1 type 3
99
24
39 measures
from which the solution can be found for
the third type of corn, then for the second,
then the first by back substitution. This
method, now known as Gaussian
elimination, would not become well known
until the early 19th Century.
Let’s get into it.
• In a great many cases, the simplest way to
describe a set of relationships uses matrix
algebra.
• Matrices are like plain numbers in many ways they can be added and subtracted, and, in some
cases, multiplied and divided and inverted.
• Some operations have unexpected results, such
as AB≠BA in general, and it is possible for A2 to
equal zero when A ≠0.
• As we shall see, matrix algebra is an important
tool for solving real-world problems.
A matrix is an array of elements - usually numbers arranged in a series of m rows and n columns.
The ORDER of a matrix specifies m and n.
12
2 10 50 0 


A   1
6 30
2 10

1
0 
60 30 0

The matrix above has order 3x5.

The elements of a matrix are specified by the ith row and jth
column. In the matrix above, the element a(3,1) is in row
three column 1. A(3,1)=-60, a(2,5)=-10, for example.
a11 a12

A  a21 a22

a31 a32
a13
a23
a33
a14
a24
a34
a15 

a25
a35

Matrix notation usually has matrix names in BOLD capital
letters.
Matrices provide a compact notation for manipulation of large
groups of numbers.
Matrix operations are easily visualized without worrying
about the large number of elements.
A few standard operations are all that are needed.
“Matlab” stands for “matrix laboratory”, and its commands
and formats are explicitly designed for matrix operations.
DEFINITIONS:
A scalar is a 1x1 matrix, or just a single number.
VECTOR: A matrix with a single row or column.
A column vector has a single column: B(n,1)
A row vector has a single row: A(1,n)
9 
 
i 
B  5
A  0.1 1 2
2 
 
20
B above has an order (5,1) and A has order (1,3).
A null matrix has all elements equal to zero. The matlab
function zeros(m,n) generates a null matrix of order m,n.
Similarly the Matlab function ones(m,n) generates a
matrix where all elements equal one.
0 0 0
zeros(2,3)  

0 0 0
1 1
 
ones(3, 2) = 1 1

1 1

A square matrix has the same number of rows and
columns.
A diagonal matrix is a square matrix with the only nonzero elements along the diagonal (i=j). We shall see that
diagonal matrices are important for scaling the values in
other matrices.
The identity matrix is a diagonal matrix (designated: I)
where the diagonal elements equal 1. Matlab generates
identity matrices with the function eye (m). The following is
an order 7,7 identity matrix:
1

0
0

I  0
0

0

0

0 0 0 0 0 0

1 0 0 0 0 0
0 1 0 0 0 0

0 0 1 0 0 0
0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

1

0
eye(5, 7) = 0

0

0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
0

0
0

0
0

The function eye(m,n) generates a non-square diagonal
matrix where the elements on the principal diagonal
alone equal 1. See above right.
A triangular matrix has zeros on one side of the diagonal.
An upper triangular matrix has all elements below the
diagonal equal to zero: 1 9 8  1 9 8 

B  0

0
A fully populated matrix has
all values non-zero.
1

5
8
B  
9
1

6
2 
2
6 4
7 6
2 4
5 3
6
7
5 

8 
4 

3
7 

43
2
0
 
22 
4 
 

2

22
4 

A sparse matrix has few nonzero elements.
1 0 0 2 


0
220
0
0

C = 
0 0 0 11


0 0 0 2 
A symmetric matrix is symmetric around the diagonal:
1
4
3
5
1


6
22 34 6 
4
S  3 22 89 2 99


18 42
5 34 2

99
42 21
1 6

A skew-symmetric matrix in which aij=-aji.

0
4
3
5
1


22 34 6 
4 0
SS   3 22 0
2 99


0
42
5 34 2

1 6 99 42 0 

Note that the diagonal elements are zero in the
skew-symmetric matrix. This is because aij
cannot equal -aji unless the value is zero when
i=j.
A matrix operation that is particularly important is
transpose. The transpose of a nxm matrix is a
mxn matrix in which the rows and columns are
switched from the original:
1 5 9 
1 2 3 4 


2 6 10


T

A  5 6 7 8 
A 
3 7 11

9 10 11 12



4 8 12
Note that the transpose of a symmetric matrix is
itself:
ST=S
The transform of a skew-symmetric matrix, S, is
-SS.
SST=-SS
The trace of a matrix, trA, is the sum of the
diagonal elements, useful in calculating various
quantities discussed later. In Matlab: function:
trace(A)
We can generate sub-matrices from larger matrices just by
grabbing appropriate sections. For example:
>> A=randint(5,7,10) % generates random integer matrix
A=
9 7 6 4 0 2 0
2 4 7 9 3 1 7
6 0 9 9 8 6 4
4 8 7 4 0 2 9
8 4 1 8 1 1 4
>> B=A(1:3,4:7) % generates a sub-matrix of A
B=
4
9
9
0
3
8
2
1
6
0
7
4
In -class exercise:
1) Generate a random integer column vector A of order
6.
2) Generate B=Transpose A
3) Generate C= Transpose B
4) Generate an order 4x4 random integer matrix, D
5) Change enough values to make it symmetric
6) Generate E= transpose D
7) Generate F by making E skew-symmetric
8) Generate G= Transpose E
9) Multiply G by -1.