This series of articles contains information and tutorials for any Matlab User.  These tutorials and guides are written with the assumption that the user has no previous knowledge of linear algebra or Matlab.  Each tutorial is broken into 4 sections and structured in the following way.

Blue: Section 1: Background and tutorial of the math and linear algebra that is used.

Green: Section 2: The MATLAB tutorial.

Orange: Section 3: Examples.

Red Section 4: Problems and learning exercises.

Vectors and Matrices what are they?

In Matlab everything is defined as either a vector or a matrix therefore it is imperative that you know what Vectors and Matrices are before you begin using Matlab.  The formal definition of a vector is something that has both a direction and a velocity. When using Matlab however, I think it is easier to think of a vector as either a row or a column of values. These values can be names, numbers, or anything you want. There are two kinds of vectors. These are row vectors and column vectors.

A Row Vector is a vector which has only One Row and can have as many columns as one wants. A row vector V with n columns is shown below.

Alternatively, a column vector is a vector which has only ONE Column and can have as many rows as one wants. A column vector V with m rows is shown below.

Matrices

A matrix is a rectangular array of quantities or expressions. I prefer to think of them as 2-D vectors. Matrices have both numerous rows and columns.

The typical representation of a matrix is m x n meaning the matrix has m rows and n columns.

The subscripts of each element refer to its position in the matrix. Matrices come in all different sizes and shapes. A matrix who has the same number of rows as columns is called a Square Matrix.

How to define Vectors and Matrices in MATLAB

Row Vectors

The use of the semicolon at the end of each line ensures that Matlab will not print off the value of V in the Command Window. If you wish to have it display this value when running, remove the semicolon.

Column Vectors

When defining column vectors in Matlab use the semicolon (;) to separate rows.

Accessing Different Vector Indices

To access different indices in a vector the following notation is used.

To access the third index or the value of 7 in both of the vector examples above one uses V(3). This tells Matlab that you are referencing the third value in the vector V.

Lets say you wish to multiply the third and fourth entry of the vector V and store that value in the first entry. You would type

V(1)=V(3)*V(4)

Matrices

Matrices in Matlab are defined by combining both row and column vectors. A 3 by 3 matrix in Matlab would be defined as follows.

MATLAB Examples

1) Define a 4 element row vector and then replace the third element by the product of the first and fourth.

2) Define a 3 element column vector and then put the value of 5 into the second element.

3) Define a 4by5 matrix and then make it so when you run the Matlab script, the element in the 3rd row and 2nd column will be printed.

Practice Problems

1) Define a 3 by 2 matrix and replace the value on the second row second column with the value on the third row first column.

2) Define a row vector with 6 elements.

3) Define a column vector with 2 elements and make sure they both print when you run Matlab (Hint: semicolon)

To start out, setup the open loop transfer function.

Next, you can choose to set up the MATLAB code in a few different ways. First make sure that both the numerator and denominator are in acceptable forms.

[1 2 3] is the same as saying

Using an example:

We can write the numerator and denominator MATLAB codes as:

>>numerator=[1];

>>denominator=[1 8];

For a more complex problem we can bypass the long and tedious expansion process and use the convolution function in MATLAB.

Here, the denominator is also represented by (s+1)*(s+3+3j)*(s+3-3j). Being three seperate parts, we can use convolution once for two of them, then use convolution again with the remaining part. Just look at the example:

conv( A , conv( B , C ) )  —> denominator = conv( [1 1], conv( [1 3+3*j], [1 3-3*j] ) );

After we define our numerator and denominator in MATLAB, we can use the root locus function then set our axis parameters as follows.

>>rlocus( numerator, denominator )

>>axis([-10 10 -10 10])

Your plot should look similar to the following for this example:

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Vector dot product rules

Another simple review of the vector dot product, for those of you that have forgotten.  The operation that involves multiplying two vectors together can be done in a few ways.  The first operation is called either the scalar product or the dot product.  One of the well known definitions looks like this:

RULE 1:

This is a scalar product that is equal to the two magnitudes multiplied together and multiplied by the cosine of the angle between them.  If  the two vectors are perpendicular to each other then the angle between them is 90 degrees, which will make the dot product equal zero.  This is an equivalent equation.

RULE 2:

When you finish your dot product, you should have a number, not a directional vector.  So if you get something like this you did something wrong: .  If you ended up with (any #) then you don’t have to completely rule out your answer.

Practice vectors

A few vectors to practice with:

Okay, now we have found our dot product by applying RULE 2 above.  We can use this value along with our individual vector magnitudes to apply RULE 1 and obtain the angle .

We can calculate the projection of the vector A onto the vector B by this relationship:

Note that this is a scalar quantity, and that we can also define the projection of B onto vector A in a similar fashion:

Performing a vector dot product in MATLAB

Perform a dot product in MATLAB like so:

>>A = [ 1 2 3];

>>B = [2 3 4];

>>dot(A,B)

Enjoy

This is a very simple post and a very simple subject, but every once in a while even the experts need to be reminded how to do simple addition and subtraction with vectors.  Let’s go ahead and specify a couple vectors that we can work with.

Vector

In MATLAB: >>A = [1 2 4];

Vector

In MATLAB: >>B = [2 3 4];

When you add vectors together, you add each individual directional component ().  Subtraction works the exact same way.  Lets go ahead and do a few, C will represent the vector that results from the addition and subtraction.

These vectors are all in 3-dimensional space with a X, Y and Z component.  The number in front of each directional component is the weight or magnitude of that particular directional component of the vector.  There it is, short and sweet

This is how you calculate the volume of an ellipsoid with the following equation.

rmax = 1;
V = 0;
step = 0.02;
A = 5;

B = 10;
C = 15;
for x= -5 : step : 5
for y = -10 : step : 10
for z = -15 : step : 15
if((x/A).^2 + (y/B).^2 + (z/C).^2) < rmax
V = V + step.^3;
end
end
end
end
disp(V)

it should spit out , approximately