Difference between revisions of "Scilab/C2/Matrix-Operations/English-timed"

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|| For example: inv of A  gives the inverse of A and spec of A gives the eigen values of  matrix A
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|| For example: inv of A  gives the inverse of A and spec of A gives the '''eigen values''' of  matrix A.
  
 
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| |See 'help spec' to see how eigenvectors can also be obtained using this command.
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| |See '''help spec''' to see how '''eigen vectors''' can also be obtained using this command.
  
 
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| |Square or cube of a square matrix A can be calculated by simply typing A square  or A cube '''respectively'''.
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| |'''Square''' or '''cube''' of a square matrix A can be calculated by simply typing '''A square''' or '''A cube '''respectively.  
  
 
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| |Please pause the tutorial now and attempt the exercise number one given with the video.
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| |Please pause the tutorial now and attempt the exercise number one given in the video.
  
 
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| | Certain special matrices can also be created in Scilab:
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| | Certain special matrices can also be created in Scilab.
  
 
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| |For example a matrix of zeros with 3 rows and 4 columns can be created using “zeros” command
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| |For example, a matrix of zeros with 3 rows and 4 columns can be created using '''zeros''' command
  
 
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| |zeros into bracket 3 comma 4  and press enter
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| |zeros into bracket 3 comma 4  and press Enter.
  
 
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| |A matrix of all ones can be created with “ones” command as follows
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| |A matrix of all ones can be created with '''ones''' command as follows:
  
 
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| |ones into bracket 2 comma 4  gives a matrix of all ones
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| |ones into bracket 2 comma 4  gives a matrix of all ones.
  
 
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| |It is easy to create an identity matrix using the  “eye” command:
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| |It is easy to create an '''identity matrix''' using the  '''eye''' command:
  
 
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| ' e y e' of 4 comma 4 gives a 4 by 4 identity matrix
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| ' e y e' of 4 comma 4 gives a 4 by 4 identity matrix.
  
 
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| |A user may need a matrix consisting of pseudo random numbers. It can be generated using the “rand” command as follows:
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| |A user may need a matrix consisting of pseudo random numbers. It can be generated using the '''rand''' command as follows:
  
 
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| |p=rand into bracket 2, 3  and press enter
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| |p=rand into bracket 2, 3  and press Enter
  
 
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| |Recall that in the Spoken Tutorial,Vector Operations, we had defined the matrix P as follows.
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| |Recall that in the Spoken Tutorial '''Vector Operations''', we had defined the matrix P as follows.
  
 
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| |P = open square bracket 1 space 2 space 3  semicolon 4 space 11 space 6  close the square bracket and press enter
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| |P = open square bracket 1 space 2 space 3  semicolon 4 space 11 space 6  close the square bracket and press Enter.
  
 
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| |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
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| |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.
  
 
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|A multiplied by x minus b
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|A multiplied by x minus b.
  
 
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| |However, one will indeed get a very small number, typically of the order of 10 raised to -16
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| |However, one will indeed get a very small number, typically of the order of 10 raised to -16.
  
 
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| |In this tutorial we have learnt
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| |In this tutorial we have learnt:
  
 
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| | To access the element of the matrix using the colon operator
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| |* To access the element of the matrix using the colon operator.
  
 
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| | Calculate the inverse of a matrix using the 'inv' command or by backslash
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| |* Calculate the inverse of a matrix using the '''inv''' command or by backslash.
  
 
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| |Calculate the derterminant of matrix using 'det' command.
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| |* Calculate the determinant of matrix using '''det''' command.
  
 
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| |Calculate eigen values of a matrix using 'spec' command.
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| |* Calculate '''eigen values''' of a matrix using '''spec''' command.
  
 
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| |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
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| |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.
  
 
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| |This spoken tutorial has been created by the Free and Open Source Software in Science and Engineering Education(FOSSEE).
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| |This spoken tutorial: has been created by the Free and Open Source Software in Science and Engineering Education(FOSSEE),
  
 
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| |This is Anuradha Amrutkar from IIT Bombay signing off.
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| |This is Anuradha Amrutkar from IIT Bombay, signing off.
  
 
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|-

Revision as of 15:22, 19 February 2015

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 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 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

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