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

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| | Welcome to the spoken tutorial on Matrix Operations.

| | Welcome to the spoken tutorial on Matrix Operations.

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| | 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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| | Access the elements of Matrix

| | Access the elements of Matrix

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| |Determine the determinant, inverse and eigen values of a matrix.

| |Determine the determinant, inverse and eigen values of a matrix.

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| |Define special matrices.

| |Define special matrices.

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| | Perform elementary row operations.

| | Perform elementary row operations.

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| | Solve the system of “linear equations”.

| | Solve the system of “linear equations”.

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|-

| | The prerequisites are

| | The prerequisites are

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| |Scilab should be installed on your system.

| |Scilab should be installed on your system.

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|-

| |''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.

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|-

| |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.

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| | Start Scilab by double-clicking on the Scilab icon present on your Desktop.

| | Start Scilab by double-clicking on the Scilab icon present on your Desktop.

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| |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.

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|-

| | Recall that in the Spoken Tutorial, 'Vector Operations',

| | Recall that in the Spoken Tutorial, 'Vector Operations',

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| | 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

| | 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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| |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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| |To access the element in the first row and second column, type E into bracket 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

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| | 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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| |For example, first row of E can be obtained using the following command: E1 = E into bracket 1 comma colon 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

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| |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.

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| |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.

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| | 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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| |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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| E2 = E of colon comma 2 colon 3 close the bracket and press enter

| E2 = E of colon comma 2 colon 3 close the bracket and press enter

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| |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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| | If the size of the matrix is not known $ (dollar ) 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 extarct the last row or column of that matrix.

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| |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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| Elast col= E into brackets colon comma dollar sign close the bracket and press enter

| Elast col= E into brackets colon comma dollar sign close the bracket and press enter

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| |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”

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| |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

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| |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

| |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

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| |Let us calculate the determinant of A by the command det of A  and press Enter.

| |Let us calculate the determinant of A by the command det of A  and press Enter.

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|| 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 eigenvalues of a matrix, the commands, “inv” and “spec” respectively, can be used.

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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

|| 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.

| |See 'help spec' to see how eigenvectors 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'''.

| |Square or cube of a square matrix A can be calculated by simply typing A square  or A cube '''respectively'''.

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| |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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| |Please pause the tutorial now and attempt the exercise number one given with the video.

| |Please pause the tutorial now and attempt the exercise number one given with the video.

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| | Certain special matrices can also be created in Scilab:

| | 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

| |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

| |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

| |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

| |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:

| |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

| ' 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:

| |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

| |p=rand into bracket 2, 3  and press enter

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| | 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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| |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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| |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.

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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

| |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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| |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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| |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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| |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

| |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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| 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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| |Rows and columns can be easily appended to matrices.

| |Rows and columns can be easily appended to matrices.

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| |For example, to append a row containing the elements [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:

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| |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

| |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

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| |The semicolon after P states that  anything after it should go to the next row. '

| |The semicolon after P states that  anything after it should go to the next row. '

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| |This is expected in the way a matrix is defined.

| |This is expected in the way a matrix is defined.

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| |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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|-

| |Matrix notations are used while solving equations.

| |Matrix notations are used while solving equations.

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|| Let us solve the following set of linear equations:

|| Let us solve the following set of linear equations:

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| |x1 + 2 x2 − x3 = 1

| |x1 + 2 x2 − x3 = 1

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| |−2 x1 − 6 x2 + 4 x3 = −2

| |−2 x1 − 6 x2 + 4 x3 = −2

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| |and − x1 − 3 x2 + 3 x3 = 1

| |and − x1 − 3 x2 + 3 x3 = 1

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| |The above set of equations can be written in the Ax = b form.

| |The above set of equations can be written in the Ax = b form.

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| | The solution is then given as inverse of A times b

| | The solution is then given as inverse of A times b

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| |Let us solve the set of equations.

| |Let us solve the set of equations.

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| |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

| |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

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| |B can be defined as b  is equal toSquare bracket 1 semicoln -2 semicolon 1 close the square bracket and press enter

| |B can be defined as b  is equal toSquare bracket 1 semicoln -2 semicolon 1 close the square bracket and press enter

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| |The solution, x, can be obtained using x = inv  of A multiplied by b

| |The solution, x, can be obtained using x = inv  of A multiplied by b

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| |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'.

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| |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.

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| |Lets do this in Scilab x  is equal to A backslash b and press enter.

| |Lets do this in Scilab x  is equal to A backslash b and press enter.

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| |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.

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| |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:

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|A multiplied by x minus b

|A multiplied by x minus b

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| |The above exercise verifies the result achieved earlier.

| |The above exercise verifies the result achieved earlier.

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| |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.

| |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.

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| |However, one will indeed get a very small number, typically of the order of 10 raised to -16

| |However, one will indeed get a very small number, typically of the order of 10 raised to -16

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| Please pause the tutorial now and attempt exercise number two given with the video.

| Please pause the tutorial now and attempt exercise number two given with the video.

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| This brings us to the end of this spoken tutorial on Matrix Operation.

| This brings us to the end of this spoken tutorial on Matrix Operation.

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| |There are many other functions in Scilab which will be covered in other spoken tutorials.

| |There are many other functions in Scilab which will be covered in other spoken tutorials.

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| Keep watching the Scilab links.

| Keep watching the Scilab links.

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|-

| |In this tutorial we have learnt

| |In this tutorial we have learnt

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| | To access the element of the matrix using the colon operator

| | 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

| | 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.

| |Calculate the derterminant of matrix using 'det' command.

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| |Calculate eigen values of a matrix using 'spec' command.

| |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

| |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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| |Solve the system of linear equations.

| |Solve the system of linear equations.

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| |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).

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|-

| |More information on the FOSSEE project could be obtained from http://fossee.in  or http://scilab.in

| |More information on the FOSSEE project could be obtained from http://fossee.in  or http://scilab.in

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|-

| | 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.  

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| | For more information, visit: http://spoken-tutorial.org/NMEICT-Intro  

| | For more information, visit: http://spoken-tutorial.org/NMEICT-Intro  

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|-

| |This is Anuradha Amrutkar from IIT Bombay signing off.

| |This is Anuradha Amrutkar from IIT Bombay signing off.

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|-

| |Thank you for joining. Goodbye.

| |Thank you for joining. Goodbye.

|}

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