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

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{| Border=1
 
{| Border=1
  
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|| Time
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|'''Time'''
  
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|| Narration
+
|'''Narration'''
  
 
|-
 
|-
  
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| 00.03
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| 00:02
  
−
| | Welcome to the spoken tutorial on Matrix Operations.
+
| | Welcome to the spoken tutorial on '''Matrix Operations'''.
  
 
|-
 
|-
  
−
|00.07
+
|00:06
  
 
| | At the end of this spoken tutorial, you will be able to:
 
| | At the end of this spoken tutorial, you will be able to:
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|-
 
|-
  
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| 00.10
+
| 00:10
  
−
| | Access the elements of Matrix
+
| |* Access the elements of Matrix
  
 
|-
 
|-
  
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| 00.13
+
| 00:13
  
−
| |Determine the determinant, inverse and eigen values of a matrix.
+
| |* Determine the determinant, inverse and eigen values of a matrix.
  
 
|-
 
|-
  
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| 00.19
+
| 00:18
  
−
| |Define special matrices.
+
| |* Define special matrices.
  
 
|-
 
|-
  
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| 00.23
+
| 00:22
  
−
| | Perform elementary row operations.
+
| |* Perform elementary row operations.
  
 
|-
 
|-
  
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| 00.25
+
| 00:25
  
−
| | Solve the system of “linear equations”.
+
| |* Solve the system of '''linear equations'''.
  
 
|-
 
|-
  
−
|00.29
+
|00:28
  
−
| | The preequisites are
+
| | The prerequisites are:
  
 
|-
 
|-
  
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| 00.31
+
| 00:30
  
 
| |Scilab should be installed on your system.
 
| |Scilab should be installed on your system.
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|-
 
|-
  
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| 00.34
+
| 00:34
  
−
| |''You should have listened to the Spoken Tutorial: Getting started with Scilab and '''Vector Operations.
+
| |You should have listened to the Spoken Tutorial: '''Getting started with Scilab''' and '''Vector Operations'''.
  
 
|-
 
|-
  
−
| 00.42
+
| 00:42
  
−
| |I am using Windows 7 operating system and Scilab 5.2.2 for demonstration.
+
| |I am using '''Windows 7''' operating system and '''Scilab 5.2.2''' for demonstration.
  
 
|-
 
|-
  
−
| 00.50
+
| 00:50
  
−
| | Start Scilab by double-clicking on the Scilab icon present on yourDesktop.
+
| | Start Scilab by double-clicking on the Scilab icon present on your Desktop.
  
 
|-
 
|-
  
−
| 00.59
+
| 00:59
  
−
| |It is suggested '''that '''the user should''' practice this tutorial in Scilab simultaneously while pausing the video at regular intervals of time.'''
+
| |It is suggested that the user should practice this tutorial in Scilab simultaneously while pausing the video at regular intervals of time.
  
 
|-
 
|-
  
−
| 01.09
+
| 01:08
  
−
| | Recall that in the Spoken Tutorial, 'Vector Operations',
+
| | Recall that in the Spoken Tutorial: '''Vector Operations''',
  
 
|-
 
|-
  
−
| 01.12
+
| 01:12
  
−
| | matrix E was defined as E = [5 19 15;8 22 36]and press enter
+
| | 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.
  
 
|-
 
|-
  
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| 01.37
+
| 01:37
  
 
| |Let us now see how to address individual elements of a matrix, separately.
 
| |Let us now see how to address individual elements of a matrix, separately.
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|-
 
|-
  
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| 01.42
+
| 01:42
  
−
| |To access the element in the first row and second column, type: E(1,2) and press enter
+
| |To access the element in the first row and second column, type E into bracket 1,2 and press enter
  
 
|-
 
|-
  
−
| 01.56
+
| 01:56
  
 
| | It is easy to extract an entire row or an entire column of a matrix in Scilab .
 
| | It is easy to extract an entire row or an entire column of a matrix in Scilab .
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|-
 
|-
  
−
| 02.03
+
| 02:03
  
−
| |For example, first row of E can be obtained using the following command: E1 = E(1,:) and press enter
+
| |For example, first row of E can be obtained using the following command: E1 = E into bracket 1 comma colon and press Enter.
  
 
|-
 
|-
  
−
| 02.24
+
| 02:23
  
−
| |The command returns all the elements of the first row in the order of their appearance in the row.
+
| |The command returns all the elements of the first row, in the order of their appearance in the row.
  
 
|-
 
|-
  
−
| 02.30
+
| 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.
+
| |'''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
+
| 02:44
  
 
| | Also, any subset of a matrix can be extracted using a colon (“:”).
 
| | Also, any subset of a matrix can be extracted using a colon (“:”).
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|-
 
|-
  
−
| 02.49
+
| 02:49
  
 
| |For example, the set of elements starting from second to third columns of E can be obtained using the following command:
 
| |For example, the set of elements starting from second to third columns of E can be obtained using the following command:
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|-
 
|-
  
−
| 03.01
+
| 03:00
  
−
| |-->E2 = E(:,2:3) close the bracket and press enter
+
| E2 = E of colon comma 2 colon 3 close the bracket and press Enter.
  
 
|-
 
|-
  
−
| 03.18
+
| 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.
 
| |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.
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|-
 
|-
  
−
|03.28
+
|03:28
  
−
| | If the size of the matrix is not known $ symbol can be used to extarct the last row or column of that matrix.
+
| | 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
+
| 03:38
  
 
| |For example to extract all rows of the last column of the matrix E, we will type
 
| |For example to extract all rows of the last column of the matrix E, we will type
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|-
 
|-
  
−
| 03.46
+
| 03:46
  
−
| |--->E last column = E(:,$) close the bracket and press enter
+
| Elast col= E into brackets colon comma dollar sign close the bracket and press Enter.
  
 
|-
 
|-
  
−
| 04.06
+
| 04:06
  
−
| |Now, let us learn how to calculate the determinant of a square matrix using the command “det”
+
| |Now, let us learn how to calculate the '''determinant''' of a '''square matrix''' using the command '''det'''.
  
 
|-
 
|-
  
−
| 04.13
+
| 04:13
  
−
| |Recall that in the Spoken Tutorial, Vector Operations, we had defined A as
+
| |Recall that in the Spoken Tutorial, '''Vector Operations''', we had defined A as
  
 
|-
 
|-
  
−
| 04.20
+
| 04:19
  
−
| |A=[1 2 - 1; - 2 - 6 4; - 1 - 3 3] close the square bracket and press enter
+
| |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
+
| 04:50
  
−
| |Let us calculate the determinant of A by the command det(A) and press Enter.
+
| |Let us calculate the determinant of A by the command '''det of A'''  and press Enter.
  
 
|-
 
|-
  
−
| 05.00
+
| 05:00
  
−
|| To calculate the inverse and the eigenvalues of a matrix, the commands, “inv” and “spec” respectively, can be used.
+
|| To calculate the '''inverse''' and the '''eigen values''' of a matrix, the commands '''inv''' and '''spec''' can be used respectively.
  
 
|-
 
|-
  
−
| 05.09
+
| 05:09
  
−
|| For example: inv(A) gives the inverse of A and spec(A) gives the eigen values of A
+
|| For example: inv of A gives the inverse of A and spec of A gives the '''eigen values''' of matrix A.
  
 
|-
 
|-
  
−
| 05.29
+
| 05:29
  
−
| |See 'help spec' to see how eigenvectors can also be obtained using this command.
+
| |See '''help spec''' to see how '''eigen vectors''' can also be obtained using this command.
  
 
|-
 
|-
  
−
| 05.35
+
| 05:35
  
−
| |Square or cube of a square matrix A can be calculated by simply typing A^2 or A^3 '''respectively'''.
+
| |'''Square''' or '''cube''' of a square matrix A can be calculated by simply typing '''A square'''  or '''A cube '''respectively.  
  
 
|-
 
|-
  
−
| 05.52
+
| 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.
 
| |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.
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|-
 
|-
  
−
| 06.05
+
| 06:05
  
−
| | '''Please pause the tutorial now and attempt exercise number one given with the video.'''
+
| |Please pause the tutorial now and attempt the exercise number one given in the video.
  
 
|-
 
|-
  
−
| 06.18
+
| 06:17
  
−
| | Certain special matrices can also be created in Scilab:
+
| | Certain special matrices can also be created in Scilab.
  
 
|-
 
|-
  
−
| 06.24
+
| 06:24
  
−
| |For example a matrix of zeros with 3 rows and 4 columns can be created using “zeros” command
+
| |For example, a matrix of zeros with 3 rows and 4 columns can be created using '''zeros''' command
  
 
|-
 
|-
  
−
| 06.36
+
| 06:36
  
−
| |zeros(3,4) and press enter
+
| |zeros into bracket 3 comma 4 and press Enter.
  
 
|-
 
|-
  
−
| 06.48
+
| 06:47
  
−
| |A matrix of all ones can be created with “ones” command as follows
+
| |A matrix of all ones can be created with '''ones''' command as follows:
  
 
|-
 
|-
  
−
| 06.53
+
| 06:53
  
−
| |ones(2,4) gives a matrix of all ones
+
| |ones into bracket 2 comma 4 gives a matrix of all ones.
  
 
|-
 
|-
  
−
| 07.02
+
| 07:01
  
−
| |It is easy to create an identity matrix using “eye” command:
+
| |It is easy to create an '''identity matrix''' using the  '''eye''' command:
  
 
|-
 
|-
  
−
| 07.07
+
| 07:07
  
−
| |eye(4,4) gives a 4 by 4 identity matrix
+
| ' e y e' of 4 comma 4 gives a 4 by 4 identity matrix.
  
 
|-
 
|-
  
−
| 07.17
+
| 07:16
  
−
| |A user may need a matrix consisting of pseudo random numbers. It can be generated using the “rand” command as follows:
+
| |A user may need a matrix consisting of pseudo random numbers. It can be generated using the '''rand''' command as follows:
  
 
|-
 
|-
  
−
| 07.25
+
| 07:25
  
−
| |p=rand(2,3) and press enter
+
| |p=rand into bracket 2, 3 and press Enter
  
 
|-
 
|-
  
−
| 07.39
+
| 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.
 
| | In linear systems, one of the important sets of operations a user carries out on matrices are the elementary row and column operations.
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|-
 
|-
  
−
| 07.55
+
| 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.
 
| |These operations involve executing row operations on a matrix to make entries below a nonzero number, zero. This can be done easily in Scilab.
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|-
 
|-
  
−
| 08.08
+
| 08:07
  
−
| |Recall that in the Spoken Tutorial,Vector Operations, we had defined the matrix P as follows.
+
| |Recall that in the Spoken Tutorial '''Vector Operations''', we had defined the matrix P as follows.
  
 
|-
 
|-
  
−
| 08.18
+
| 08:17
  
−
| |P = [1 2 3;4 11 6]close the square bracket and press enter
+
| |P = open square bracket 1 space 2 space 3 semicolon 4 space 11 space 6 close the square bracket and press Enter.
  
 
|-
 
|-
  
−
| 08.34
+
| 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.
 
| |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.
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|-
 
|-
  
−
| 08.45
+
| 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:
 
| |The operation can be executed by multiplying the first row by 4 and subtracting it from the second row as in the following command:
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|-
 
|-
  
−
| 08.57
+
| 08:56
  
−
| |P(2,:) = P(2,:) - 4*P(1,:) and press enter
+
| |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.
  
 
|-
 
|-
  
−
| 09.29
+
| 09:28
  
 
| The procedure can be extended to larger systems and to other forms of elementary column operations.
 
| The procedure can be extended to larger systems and to other forms of elementary column operations.
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|-
 
|-
  
−
| 09.35
+
| 09:35
  
 
| |Rows and columns can be easily appended to matrices.
 
| |Rows and columns can be easily appended to matrices.
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|-
 
|-
  
−
| 09.39
+
| 09:39
  
−
| |For example, to append a row containing [5 5 -2] to P, the following command is used:
+
| |For example, to append a row containing the elements [5 5 -2] to P, the following command is used:
  
 
|-
 
|-
  
−
|09.49
+
|09:48
  
−
| |T = [P; [5 5 -2]] close both the square bracket and press enter
+
| |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.
  
 
|-
 
|-
  
−
| 10.15
+
| 10:14
  
−
| |The semicolon after P states that the anything after it should go to the next row. '
+
| |The semicolon after P states that anything after it should go to the next row.  
  
 
|-
 
|-
  
−
| 10.20
+
| 10:20
  
 
| |This is expected in the way a matrix is defined.
 
| |This is expected in the way a matrix is defined.
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|-
 
|-
  
−
| 10.24
+
| 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.
 
| |As an exercise, please pause here and check if the brackets around the new row, in the command just executed, are really required.
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|-
 
|-
  
−
| 10.34
+
| 10:34
  
 
| |Matrix notations are used while solving equations.
 
| |Matrix notations are used while solving equations.
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|-
 
|-
  
−
| 10.41
+
| 10:40
  
 
|| Let us solve the following set of linear equations:
 
|| Let us solve the following set of linear equations:
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|-
 
|-
  
−
| 10.44
+
| 10:44
  
 
| |x1 + 2 x2 − x3 = 1
 
| |x1 + 2 x2 − x3 = 1
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|-
 
|-
  
−
| 10.48
+
| 10:48
  
−
| |− 2 x1 − 6 x2 + 4 x3 = − 2
+
| |−2 x1 − 6 x2 + 4 x3 = −2
  
 
|-
 
|-
  
−
| 10.54
+
| 10:54
  
−
| |− x1 − 3 x2 + 3 x3 = 1
+
| |and − x1 − 3 x2 + 3 x3 = 1
  
 
|-
 
|-
  
−
| 11.00
+
| 11:00
  
−
| |The above set of equations can be written in the Ax = b form.
+
| |The above set of equations can be written in Ax = b form.
  
 
|-
 
|-
  
−
| 11.05
+
| 11:05
  
−
| | The solution is then given as inverse of A times b
+
| | The solution is then given as inverse of A times b.
  
 
|-
 
|-
  
−
| 11.11
+
| 11:11
  
 
| |Let us solve the set of equations.
 
| |Let us solve the set of equations.
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|-
 
|-
  
−
| 11.15
+
| 11:15
  
−
| |A is defined as A = [1 2 -1;-2 -6 4;-1 -3 3] close the square bracket and press enter
+
| |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
+
| 11:46
  
−
| |B can be defined as b = [1;-2;1]close the square bracket and press enter
+
| |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
+
| 12:04
  
−
| |The solution, x, can be obtained using x = inv(A)*b
+
| |The solution, x, can be obtained by using x = inv of A multiplied by b.
  
 
|-
 
|-
  
−
| 12.20
+
| 12:19
  
−
| |It is worth noting that it is a small letter 'i' in the command, 'inv'.
+
| |It is worth noting that it is a small letter 'i' in the command, '''inv'''.
  
 
|-
 
|-
  
−
| 12.27
+
| 12:26
  
−
| |Alternatively, the same result can be achieved using a backslash operation in Scilab.
+
| |Alternatively, the same result can be achieved using a '''backslash operation''' in Scilab.
  
 
|-
 
|-
  
−
| 12.33
+
| 12:33
  
−
| |Lets do this in Scilab x = A\b and press enter.
+
| |Let's do this in Scilab: x is equal to A backslash b and press Enter.
  
 
|-
 
|-
  
−
| 12.45
+
| 12:44
  
−
| |It gives the same result. Type "help backslash" and "help inv" in Scilab to know more about individual advantages and disadvantages.
+
| |It gives the same result. Type '''help backslash''' and '''help inv''' in Scilab to know more about individual advantages and disadvantages.
  
 
|-
 
|-
  
−
| 12.55
+
| 12:55
  
 
| |The integrity of the solution can be verified by back substitution, that is, by calculating Ax-b:
 
| |The integrity of the solution can be verified by back substitution, that is, by calculating Ax-b:
Line 471: Line 469:
 
|-
 
|-
  
−
|13.05
+
|13:05
  
−
|A*x-b
+
|A multiplied by x minus b.
  
 
|-
 
|-
  
−
| 13.10
+
| 13:10
  
 
| |The above exercise verifies the result achieved earlier.
 
| |The above exercise verifies the result achieved earlier.
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|-
 
|-
  
−
| 13.14
+
| 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
+
| |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.
  
 
|-
 
|-
  
−
| 13.35
+
| 13:27
  
−
| '''Please pause the tutorial now and attempt exercise number two given with the video.'''
+
| |However, one will indeed get a very small number, typically of the order of 10 raised to -16.
  
 
|-
 
|-
  
−
| 13.49
+
| 13:34
  
−
| '''This brings us to the end of this spoken tutorial on Matrix Operation.'''
+
| Please pause the tutorial now and attempt exercise number two given in the video.
  
 
|-
 
|-
  
−
| 13.54
+
| 13:49
  
−
| |'''There are many other functions in Scilab which will be covered in other spoken tutorials.
+
| This brings us to the end of this spoken tutorial on '''MatrixOperation'''.                                                                                                                      
  
 
|-
 
|-
  
−
| 13.59
+
| 13:53
  
−
| Keep watching the Scilab links.'''
+
| |There are many other functions in Scilab which will be covered in other spoken tutorials.
  
 
|-
 
|-
  
−
| 14.02
+
| 13:59
  
−
| |'''In this tutorial we have learnt'''
+
| Keep watching the Scilab links.
  
 
|-
 
|-
  
−
| 14.04
+
| 14:02
  
−
| | '''To access the element of the matrix using the colon operator'''
+
| |In this tutorial we have learnt:
  
 
|-
 
|-
  
−
| 14.08
+
| 14:04
  
−
| | '''Calculate the inverse of a matrix using the 'inv' command or by backslash'''
+
| |* To access the element of the matrix using the colon operator.
  
 
|-
 
|-
  
−
| 14.14
+
| 14:07
  
−
| |'''Calculate the derterminant of matrix using 'det' command.'''
+
| |* Calculate the inverse of a matrix using the '''inv''' command or by backslash.
  
 
|-
 
|-
  
−
| 14.19
+
| 14:14
  
−
| | '''Calculate eigen values of a matrix using 'spec' command.'''
+
| |* Calculate the determinant of matrix using '''det''' command.
  
 
|-
 
|-
  
−
| 14.24
+
| 14:18
  
−
| | '''Define a matrix having all the elements one, Null Matrix,
+
| |* Calculate '''eigen values''' of a matrix using '''spec''' command.
  
 
|-
 
|-
  
−
| 14.29
+
| 14:23
  
−
| Identity matrix and a matrix with random elements by using functions ones(), zeros(), eye(), rand() respectively'''
+
| |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
+
| 14:39
  
−
| | '''Solve the system of linear equations.'''
+
| |Solve the system of linear equations.
  
 
|-
 
|-
  
−
| 14.43
+
| 14:42
  
−
| | '''This spoken tutorial has been created by the Free and Open Source Software in Science and Engineering Education(FOSSEE). '''
+
| |This spoken tutorial: has been created by the Free and Open Source Software in Science and Engineering Education(FOSSEE),
  
 
|-
 
|-
  
−
| 14.51
+
| 14:51
  
−
| | '''More information on the FOSSEE project could be obtained from [http://fossee.in/ http://fossee.in] or [http://scilab.in/ ] '''
+
| |More information on the FOSSEE project could be obtained from http://fossee.in or http://scilab.in
  
 
|-
 
|-
  
−
| 14.59
+
| 14:58
  
−
| | '''Supported by the National Mission on Eduction through ICT, MHRD, Government of India. '''
+
| | Supported by the National Mission on Eduction through ICT, MHRD, Government of India.  
  
 
|-
 
|-
  
−
| 15.05
+
| 15:05
  
−
| | '''For more information, visit: [http://spoken-tutorial.org/NMEICT-Intro http://spoken-tutorial.org/NMEICT-Intro] '''
+
| | For more information, visit: http://spoken-tutorial.org/NMEICT-Intro  
  
 
|-
 
|-
  
−
| 15.15
+
| 15:14
  
−
| |'''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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|15:18
  
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| |'''Thank you for joining. Goodbye.'''
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| |Thank you for joining. Goodbye.
  
 
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Latest revision as of 16:53, 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 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

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