Welcome to this spoken tutorial on Matrix Operations using Scilab.
To practice this tutorial, you need to have Scilab installed on your system. Open the
scilab console window.
Let us start by defining a vector. This can be done in two ways: first one is using
spaces
-->p = [1 2 3]
p =
1. 2. 3.
or using commas.
-->q = [2, 3, 4]
q =
2. 3. 4.
We can find the length of a vector as follows:
-->length(p)
ans =
3.
We can work with vectors, just as we do in Maths classes. We can add them:
-->p + q
ans =
3. 5. 7.
We can subtract
-->q - p
ans =
1. 1. 1.
and so on.
Transpose of a vector can be found using apostrophe.
-->p'
ans =
1.
2.
3.
This is also known as a prime or a single quote. On my keyboard, it is next to the
enter or return key.
We can calculate p-transpose (times)* q:
-->p'*q
ans =
2. 3. 4.
4. 6. 8.
6. 9. 12.
The command p times q-transpose gives a scalar:
-->p*q'
ans =
20.
I suggest that you pause at this point and check these calculations. I would want
you to do this at regular intervals.
When defining matrices, the semicolon is used for defining the next row. For
example, define a 2 by 3 matrix P by typing
-->P = [1 2 3;4 5 6]
P =
1. 2. 3.
4. 5. 6.
Note that this P is in upper case. In Scilab, upper case variables are different from
lower case variables. Would you want to check what small p has at this point?
The size of a matrix can be obtained by using the size command
-->size(P)
ans =
2. 3.
answer is 2 , 3
We can assign these values to variables as follows:
-->[row, col] = size(P)
col =
3.
row =
2.
The transpose command works for matrices as well:
-->P'
ans =
1. 4.
2. 5.
3. 6.
as given here
Let us define another 2 by 3 matrix Q:
-->Q = [1 5 3;2 4 8]
Q =
1. 5. 3.
2. 4. 8.
Let us also recall P once more:
We can carry out calculations involving P and Q, just as we do in mathematics.
For example, let us calculate:
-->E = 2*P + 3*Q
E =
5. 19. 15.
14. 22. 36.
You may want to verify whether these calculations are correct.
We can address the elements of matrices easily. For example, if we want to access
the 2,2 element, we give:
-->E(2,2)
ans =
22
It is easy to address rows and columns of matrices. For example, the first row of Q
is obtained using the following command:
-->Efr = E(1,:)
Efr =
5. 19. 15.
You see that we get the first row.
The semicolon operator says, take everything. Here, it says that we have to take all
columns, namely, three columns. Let us recall E. The second to third columns of E
are obtained using
-->Est = E(:,2,3)
I made a mistake
so 2 to 3 is given by this colon operator
-->Est = E(:,2:3)
Est =
19. 15.
22. 36.
Ok you can see that. Let recall what E is.
You can see that.
The first colon says take everything.
2 to 3 says take from 2 to 3 onwards
I have a question for you. What will E(:,:) give?
I will not give this command.
I would want you to try this out.
I would want you to explain why it happens.
One of the important operations carried out on matrices is elementary row
operations. In this, we have to make entries below a number to be zero. This can
be done easily in Scilab.
Let us recall P.
Suppose that we want to make the (2,1) entry zero by elementary row operation.
That is, we would want to multiply the first row by 4 and subtract it from the second
row:
-->P(2,:) = P(2,:) - 4*P(1,:)
P =
1. 2. 3.
0. - 3. - 6.
Would you want to pause and check out this calculation?
This procedure can be extended to larger systems and to elementary column
operations.
We can easily append rows and columns to matrices. For example, if we want to
append a row containing [5 5 -2], we do the following:
-->T = [P; (go to the next row) [5 5 -2]]
(this row is over this operation is over)
T =
1. 2. 3.
0. - 3. - 6.
5. 5. - 2.
The semicolon after P says that the numbers coming after should go to the next row.
We use matrix notation while solving equations. Suppose that we want to solve this
matrix equation:
<math>
x_1 + 2x_2 - x_3 = 1
</math>
<math>
-2x_1 - 6x_2 + 4x_3 = -2
</math>
<math>
-x_1 - 3x_2 + 3x_3 = 1
</math>
This can be written as follows Ax = b
The solution is given as Inverse A times b
Lets verify whether this is correct
A is given as
-->A = [1 2 -1;-2 -6 4;-1 -3 3]
A =
1. 2. - 1.
- 2. - 6. 4.
- 1. - 3. 3.
-->b = [1;-2;1]
b =
1.
- 2.
1.
This is the same system we saw here
Let get back here
We can calculate x using
-->x = inv(A)*b
x =
- 1.
2.
2.
we get this result
We can also solve it in scilab with A reverse slash b
It is as if this A is taken from this side over here
so it comes to the left of b and because A divides b you put a slash in this manner
Let us do it in Scilab
-->x = A\b
x =
- 1.
2.
2.
We get the same result
The first method involves two steps: calculation of inverse of A and then
multiplication with b.
A\b just tells Scilab to solve this system. This is done in one step. As a result, A\b is
generally more efficient.
We can verify the correctness of the solution by back substitution, that is, by
calculating Ax-b:
-->A*x-b
ans =
0.
0.
0.
We see that the solution is indeed correct. It is possible that in some systems we
may not get zero exactly. But we will get a number close to zero, say, of the order
of 10^{-16} or something like that.
For square matrices, we can calculate the determinants:
Let us recall A first:
Lets calculate the determinant of A
-->det(A)
ans =
- 2.
We can calculate the square or cube of a square matrix A by simply typing A^2 or
A^3. Here we use the carat symbol. In my keyboard, it is obtained by shift 6.
One can calculate the eigenvalues of a square matrix by the command spec, which
is a short form for spectrum:
as i do here
-->spec(A)
ans =
- 3.7448261
0.3959319
1.3488942
It is possible to calculate the eigenvectors also using the spec command. You can
find the syntax for this by giving this command
-->help spec
I will not give this command, however.
Using Scilab we can create special matrices:
For example we can create a matrix of zeros with 3 rows and 4 columns using this
command
-->zeros(3,4)
ans =
0. 0. 0. 0.
0. 0. 0. 0.
0. 0. 0. 0.
A matrix of all ones can be created with ones command as follows
-->ones(2,4)
ans =
1. 1. 1. 1.
1. 1. 1. 1.
It is easy to create identity matrices as well:
-->eye(4,4)
ans =
1. 0. 0. 0.
0. 1. 0. 0.
0. 0. 1. 0.
0. 0. 0. 1.
This is a 4 by 4 identity matrix.
We often need matrices consisting of pseudo random numbers. These can be
generated by using the rand command as follows
-->p=rand(2,3)
p =
0.2113249 0.0002211 0.6653811
0.7560439 0.3303271 0.6283918
See help rand for options regarding the distributions and seeds.
This brings us to the end of spoken tutorial on Matrix Operations using Scilab.
We also have several other spoken tutorial on Scilab at this time.
These are listed here.
Scilab Effort in India is co-ordinated through this website.
scilab.in
There are some interesting projects one of them is the Textbook project
That codes worked out examples of standard textbooks using scilab
The link project allows users to link known scilab documents and to rank them
We also help organize Scilab Workshops
We have two mailing lists one for announce and another one for discuss.
We invite your participation in all our activities
Lets get back to spoken tutorials
The spoken part will be available in various Indian Languages as well.
These are available at spoken-tutorial.org
These tutorial form a part of Level 0 training in Scilab.
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government of India. More information on the same is available at this website
http://spoken-tutorial.org/NMEICT-Intro.
Thanks for joining us. This is Kannan Moudgalya signing off. Goodbye.
