Difference between revisions of "Scilab/C2/Matrix-Operations/English-timed"
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Revision as of 14:49, 6 March 2014
| 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 preequisites 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 extarct 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 eigenvalues of a matrix, the commands, “inv” and “spec” respectively, can be used. |
| 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 eigenvectors 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 with 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 bracket 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 the 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 toSquare bracket 1 semicoln -2 semicolon 1 close the square bracket and press enter |
| 12.04 | The solution, x, can be obtained 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 | Lets 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 with the video. |
| 13.49 | This brings us to the end of this spoken tutorial on Matrix Operation. |
| 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 derterminant 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
