Scilab/C2/Matrix-Operations/English-timed

Time	Narration
00:02	Welcome to the spoken tutorial on Matrix Operations.
00:06	At the end of this spoken tutorial, you will be able to:
00:10	* Access the elements of Matrix
00:13	* Determine the determinant, inverse and eigen values of a matrix.
00:18	* Define special matrices.
00:22	* Perform elementary row operations.
00:25	* Solve the system of linear equations.
00:28	The prerequisites are:
00:30	Scilab should be installed on your system.
00:34	You should have listened to the Spoken Tutorial: Getting started with Scilab and Vector Operations.
00:42	I am using Windows 7 operating system and Scilab 5.2.2 for demonstration.
00:50	Start Scilab by double-clicking on the Scilab icon present on your Desktop.
00:59	It is suggested that the user should practice this tutorial in Scilab simultaneously while pausing the video at regular intervals of time.
01:08	Recall that in the Spoken Tutorial: Vector Operations,
01:12	matrix E was defined as E is equal to open square bracket 5 space 19 space 15 semicolon 8 space 22 space 36 close the square bracket and press Enter.
01:37	Let us now see how to address individual elements of a matrix, separately.
01:42	To access the element in the first row and second column, type E into bracket 1,2 and press enter
01:56	It is easy to extract an entire row or an entire column of a matrix in Scilab .
02:03	For example, first row of E can be obtained using the following command: E1 = E into bracket 1 comma colon and press Enter.
02:23	The command returns all the elements of the first row, in the order of their appearance in the row.
02:30	Colon, when used alone, refers to all the elements of row or column, depending upon whether it appears as a first or a second entry respectively inside the bracket.
02:44	Also, any subset of a matrix can be extracted using a colon (“:”).
02:49	For example, the set of elements starting from second to third columns of E can be obtained using the following command:
03:00	E2 = E of colon comma 2 colon 3 close the bracket and press Enter.
03:18	In the above, the second entry in the bracket, that is, "2 colon 3" makes a reference to elements from column 2 to column 3.
03:28	If the size of the matrix is not known, $ (dollar ) symbol can be used to extract the last row or column of that matrix.
03:38	For example to extract all rows of the last column of the matrix E, we will type
03:46	Elast col= E into brackets colon comma dollar sign close the bracket and press Enter.
04:06	Now, let us learn how to calculate the determinant of a square matrix using the command det.
04:13	Recall that in the Spoken Tutorial, Vector Operations, we had defined A as
04:19	A = open square bracket 1 space 2 space minus 1 semicolon -2 space - 6 space 4 semicolon -1 space -3 space 3 close the square bracket and press Enter.
04:50	Let us calculate the determinant of A by the command det of A and press Enter.
05:00	To calculate the inverse and the eigen values of a matrix, the commands inv and spec can be used respectively.
05:09	For example: inv of A gives the inverse of A and spec of A gives the eigen values of matrix A.
05:29	See help spec to see how eigen vectors can also be obtained using this command.
05:35	Square or cube of a square matrix A can be calculated by simply typing A square or A cube respectively.
05:52	A caret symbol is used to raise a matrix to power, like in ordinary arithmetic operations. In our keyboard, it is obtained by pressing shift+6.
06:05	Please pause the tutorial now and attempt the exercise number one given in the video.
06:17	Certain special matrices can also be created in Scilab.
06:24	For example, a matrix of zeros with 3 rows and 4 columns can be created using zeros command
06:36	zeros into bracket 3 comma 4 and press Enter.
06:47	A matrix of all ones can be created with ones command as follows:
06:53	ones into bracket 2 comma 4 gives a matrix of all ones.
07:01	It is easy to create an identity matrix using the eye command:
07:07	' e y e' of 4 comma 4 gives a 4 by 4 identity matrix.
07:16	A user may need a matrix consisting of pseudo random numbers. It can be generated using the rand command as follows:
07:25	p=rand into bracket 2, 3 and press Enter
07:39	In linear systems, one of the important sets of operations a user carries out on matrices are the elementary row and column operations.
07:55	These operations involve executing row operations on a matrix to make entries below a nonzero number, zero. This can be done easily in Scilab.
08:07	Recall that in the Spoken Tutorial Vector Operations, we had defined the matrix P as follows.
08:17	P = open square bracket 1 space 2 space 3 semicolon 4 space 11 space 6 close the square bracket and press Enter.
08:33	Let us consider an example where the element in the second row, first column is to be transformed to zero using elementary row and column operation.
08:44	The operation can be executed by multiplying the first row by 4 and subtracting it from the second row as in the following command:
08:56	P into bracket 2 comma colon is equal to P into bracket 2 comma colon minus 4 multiplied by P into bracket 1 comma colon and press Enter.
09:28	The procedure can be extended to larger systems and to other forms of elementary column operations.
09:35	Rows and columns can be easily appended to matrices.
09:39	For example, to append a row containing the elements [5 5 -2] to P, the following command is used:
09:48	T = open square bracket P semicolon, open another square bracket write down the elements 5 5 -2 close both the square brackets and press Enter.
10:14	The semicolon after P states that anything after it should go to the next row.
10:20	This is expected in the way a matrix is defined.
10:24	As an exercise, please pause here and check if the brackets around the new row, in the command just executed, are really required.
10:34	Matrix notations are used while solving equations.
10:40	Let us solve the following set of linear equations:
10:44	x1 + 2 x2 − x3 = 1
10:48	−2 x1 − 6 x2 + 4 x3 = −2
10:54	and − x1 − 3 x2 + 3 x3 = 1
11:00	The above set of equations can be written in Ax = b form.
11:05	The solution is then given as inverse of A times b.
11:11	Let us solve the set of equations.
11:15	A is defined as A = open square bracket 1 space 2 space -1 semicolon -2 space -6 space 4 semicolon -1 space -3 space 3 close the square bracket and press Enter.
11:46	B can be defined as b is equal to square bracket 1 semicolon -2 semicolon 1 close the square bracket and press Enter.
12:04	The solution, x, can be obtained by using x = inv of A multiplied by b.
12:19	It is worth noting that it is a small letter 'i' in the command, inv.
12:26	Alternatively, the same result can be achieved using a backslash operation in Scilab.
12:33	Let's do this in Scilab: x is equal to A backslash b and press Enter.
12:44	It gives the same result. Type help backslash and help inv in Scilab to know more about individual advantages and disadvantages.
12:55	The integrity of the solution can be verified by back substitution, that is, by calculating Ax-b:
13:05	A multiplied by x minus b.
13:10	The above exercise verifies the result achieved earlier.
13:14	It is possible that in some systems the above verification exercise may not yield a matrix with exact zeros as its elements due to intermediate floating point operations.
13:27	However, one will indeed get a very small number, typically of the order of 10 raised to -16.
13:34	Please pause the tutorial now and attempt exercise number two given in the video.
13:49	This brings us to the end of this spoken tutorial on MatrixOperation.
13:53	There are many other functions in Scilab which will be covered in other spoken tutorials.
13:59	Keep watching the Scilab links.
14:02	In this tutorial we have learnt:
14:04	* To access the element of the matrix using the colon operator.
14:07	* Calculate the inverse of a matrix using the inv command or by backslash.
14:14	* Calculate the determinant of matrix using det command.
14:18	* Calculate eigen values of a matrix using spec command.
14:23	Define a matrix having all the elements one, Null Matrix, Identity matrix and a matrix with random elements by using functions ones(), zeros(), eye(), rand() respectively.
14:39	Solve the system of linear equations.
14:42	This spoken tutorial: has been created by the Free and Open Source Software in Science and Engineering Education(FOSSEE),
14:51	More information on the FOSSEE project could be obtained from http://fossee.in or http://scilab.in
14:58	Supported by the National Mission on Eduction through ICT, MHRD, Government of India.
15:05	For more information, visit: http://spoken-tutorial.org/NMEICT-Intro
15:14	This is Anuradha Amrutkar from IIT Bombay, signing off.
15:18	Thank you for joining. Goodbye.

Contributors and Content Editors

Gaurav, Jyotisolanki, PoojaMoolya, Ranjana, Sandhya.np14, Sneha