Transcript Chapter 4 Review

Chapter 4 Review:
Manipulating Matrices
Introduction to MATLAB 7
Engineering 161
Defining Matrices I
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In MATLAB, a matrix can be defined by typing a list
of numbers enclosed by square brackets.
A matrix can also be defined by listing each row on a
separate line.
When you get more comfortable with MATLAB you
find other short hand ways to enter matrices, row or
column vectors.
Use … to extend a line if needed to enter large
matrices or long row vectors.
Defining Matrices II
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Suppose s = [1.5, 2.2, 1.7, -4.1], then s(3) = 1.7,
and so on.
You can use the word end to identify the final
element in a row or column, so s(end) = -4.1
A = [ ] is the empty matrix with length equal to 0
For 2 dimensional matrices you can identify an
element by either specifying both the row and
column or using a single index number where the
elements are listed by column.
Problems with Two Variables I
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So far we have only considered problems of
the form y = f(x), that is, problems of a
single variable. Often in engineering we have
to deal with problems of two variables, where
z = g(x,y).
MATLAB has a built in function called
meshgrid to help accomplish this.
Consider the following example;
Problems with Two Variables II
x = 0:5;
y = 1:1:3;
And we want to compute, for example,
z = x2 +xy + y2
If we write the MATLAB statement
z = x.^2 + x.*y + y.^2
We will get an error message. Why??
Problems with Two Variables III
Now consider;
x = 0:5;
y = 1:1:3;
[X,Y] = meshgrid(x,y);
z = X.^2 + X.*Y + Y.^2
The new matrices X and Y created by the
meshgrid function are now of the same size
and the expression for z makes sense.
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Problems with Two Variable IV
The new matrices look like;
X=0 1 2 3 4 5
0 1 2 3 4 5
0 1 2 3 4 5
z = X.^2 + X.*Y + Y.^2
Y= 1 1 1 1 1
2 2 2 2 2
3 3 3 3 3
Problems with Two Variable V
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When we perform the computation as given in the previous slides, all
points will be calculated using one MATLAB statement. Let’s look at
another example;
Suppose we want to calculate d= 1/2gt2 for both the moon and earth
for 0< t < 10 seconds. Here g and t are vectors, g = [2.4, 9.8] and for
t = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. Consider the following MATLAB
statements;
t = 0:10;
g = [ 2.4, 9.8];
[T, G] = meshgrid(t,g);
d = 0.5*G.*T.^2;
results = [t’, d’]
d would be an 2x11 matrix, results for both the earth and moon
Special Matrices I
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MATLAB contains a number of built in functions that
generate special matrices. You’ll find these matrices
very useful as you advance with you MATLAB skills.
Matrix of zeros: A = zeros(3) or B = zeros(2,4)
Matrix of ones:
A = ones(2) or B = ones(2,5)
Magic matrices: C = magic(4) creates a matrix
called a Magic Square that has special properties.
Special Matrices II
Diagonal matrices: Use the function diag(A) in one of two
ways;
If A is mxn matrix, B = diag(A) returns B as a column vector of
the diagonal elements of A. B = diag(A, 1) would produce the
elements of the diagonal above the main diagonal, diag(A, -1),
below the main diagonal.
If A is a row matrix, B = diag(A) produces a matrix with the
elements of A form the diagonal of B, with zeros everywhere
else.
Chapter 4 Assignments
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Chapter 4 assignments let you practice
working with problems with two
variables and matrices. They are,
4.1, 4.8, 4.12.