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diff --git a/matrices/script.rst b/matrices/script.rst index 30a2856..a337605 100644 --- a/matrices/script.rst +++ b/matrices/script.rst @@ -31,36 +31,44 @@ Language Reviewer : Bhanukiran Checklist OK? : <11-11-2010, Anand, OK> [2010-10-05] -.. #[punch: please mark the exercises, using the syntax we decided upon.] ======== Matrices ======== -{{{ show the welcome slide }}} -Welcome to the spoken tutorial on Matrices. +{{{ Show the first slide containing title, name of the production +team along with the logo of MHRD }}} -{{{ switch to next slide, outline slide }}} +Hello friends and welcome to the tutorial on 'Matrices'. -In this tutorial we will learn about matrices, creating matrices using -direct data, converting a list and matrix operations. Finding -inverse of a matrix, determinant of a matrix, eigen values and eigen -vectors of a matrix, norm and singular value decomposition of -matrices. +{{{ switch to slide with objectives }}} -{{{ creating a matrix }}} +At the end of this tutorial, you will be able to, + + 1. Create matrices using data. + #. Create matrices from lists. + #. Do basic matrix operations like addition,multiplication. + #. Perform operations to find out the -- + - inverse of a matrix + - determinant of a matrix + - eigen values and eigen vectors of a matrix + - norm of a matrix + - singular value decomposition of a matrix. + +{{{ Switch to the pre-requisite slide }}} + +Before beginning this tutorial,we would suggest you to complete the +tutorial on "Getting started with lists", "Getting started with arrays", +"Accessing parts of arrays". All matrix operations are done using arrays. Thus all the operations on arrays are valid on matrices also. A matrix may be created as, :: + ipython -pylab m1 = array([1,2,3,4]) - -.. #[Puneeth: don't use ``matrix``. Use ``array``. The whole script will -.. have to be fixed.] - -Using the method ``shape`` we can find out the shape or size of the +Using the method ``shape``, we can find out the shape or size of the matrix, :: @@ -75,12 +83,12 @@ A list can be converted to a matrix as follows, l1 = [[1,2,3,4],[5,6,7,8]] m2 = array(l1) -{{{ switch to next slide, exercise 1}}} +Pause the video here, try out the following exercise and resume the video. -Pause here and create a two dimensional matrix m3 of order 2 by 4 with -elements 5, 6, 7, 8, 9, 10, 11, 12. +{{{ switch to slide, exercise 1}}} -{{{ switch to next slide, solution }}} +Create a two dimensional matrix m3 of order 2 by 4 with +elements 5, 6, 7, 8, 9, 10, 11, 12. m3 can be created as, :: @@ -89,22 +97,21 @@ m3 can be created as, {{{ switch to next slide, matrix operations }}} -We can do matrix addition and subtraction as, +We can do matrix addition and subtraction easily. :: m3 + m2 -does element by element addition, thus matrix addition. +m3+m2 does element by element addition, that is matrix addition. Similarly, :: m3 - m2 -it does matrix subtraction, that is element by element -subtraction. Now let us try, - -{{{ Switch to next slide, Matrix multiplication }}} +Similarly,m3-m2 does matrix subtraction, that is element by element +subtraction. +Now let us try,matrix multiplication :: m3 * m2 @@ -120,8 +127,6 @@ And matrix multiplication in matrices are done using the function ``dot()`` but due to size mismatch the multiplication could not be done and it returned an error, -{{{ switch to next slide, Matrix multiplication (cont'd) }}} - Now let us see an example for matrix multiplication. For doing matrix multiplication we need to have two matrices of the order n by m and m by r and the resulting matrix will be of the order n by r. Thus let us @@ -130,7 +135,7 @@ first create two matrices which are compatible for multiplication. m1.shape -matrix m1 is of the shape one by four, let us create another one of +matrix m1 is of the shape one by four, let us create another one, of the order four by two, :: @@ -141,15 +146,14 @@ thus the function ``dot()`` can be used for matrix multiplication. {{{ switch to next slide, recall from arrays }}} -As we already saw in arrays, the functions ``identity()`` which -creates an identity matrix of the order n by n, ``zeros()`` which -creates a matrix of the order m by n with all zeros, ``zeros_like()`` -which creates a matrix with zeros with the shape of the matrix passed, -``ones()`` which creates a matrix of order m by n with all ones, -``ones_like()`` which creates a matrix with ones with the shape of the -matrix passed. These functions can also be used with matrices. - -{{{ switch to next slide, more matrix operations }}} +As we already learnt in arrays, the function ``identity()`` which +creates an identity matrix of the order n by n, the function ``zeros()`` +which creates a matrix of the order m by n with all zeros, the function +``zeros_like()`` which creates a matrix with zeros with the shape of +the matrix passed, the function ``ones()`` which creates a matrix of +order m by n with all ones, the function ``ones_like()`` which creates a +matrix with ones with the shape of the matrix passed; all these +functions can also be used with matrices. To find out the transpose of a matrix we can do, :: @@ -161,8 +165,6 @@ Matrix name dot capital T will give the transpose of a matrix {{{ switch to next slide, Frobenius norm of inverse of matrix }}} -.. #[punch: arange has not been introduced.] - Now let us try to find out the Frobenius norm of inverse of a 4 by 4 matrix, the matrix being, :: @@ -170,23 +172,17 @@ matrix, the matrix being, m5 = arange(1,17).reshape(4,4) print m5 -The inverse of a matrix A, A raise to minus one is also called the -reciprocal matrix such that A multiplied by A inverse will give 1. The +The inverse of a matrix A, A raise to minus one, is also called the +reciprocal matrix, such that A multiplied by A inverse will give 1. The Frobenius norm of a matrix is defined as square root of sum of squares -of elements in the matrix. Pause here and try to solve the problem -yourself, the inverse of a matrix can be found using the function -``inv(A)``. +of elements in the matrix. The inverse of a matrix can be found using the +function ``inv(A)``. And here is the solution, first let us find the inverse of matrix m5. :: im5 = inv(m5) -.. #[punch: we don't need to show this way of calculating the norm, do -.. we? even if we do, we should show it in the "array style". -.. something like: -.. sqrt(sum(each * each))] - And the Frobenius norm of the matrix ``im5`` can be found out as, :: @@ -196,11 +192,10 @@ And the Frobenius norm of the matrix ``im5`` can be found out as, print sqrt(sum) {{{ switch to next slide, infinity norm }}} -.. #[punch: similarly for this section.] -Now try to find out the infinity norm of the matrix im5. The infinity -norm of a matrix is defined as the maximum value of sum of the -absolute of elements in each row. Pause here and try to solve the +Now let us move on to find out the infinity norm of the matrix im5. +The infinity norm of a matrix is defined as the maximum value of sum of +the absolute of elements in each row. Pause here and try to solve the problem yourself. @@ -229,42 +224,39 @@ And to find out the Infinity norm of the matrix im5, we do, norm(im5,ord=inf) -This is easier when compared to the code we wrote. Do ``norm`` -question mark to read up more about ord and the possible type of norms +This is easier when compared to the code we wrote. Read the documentation +of ``norm`` to read up more about ord and the possible type of norms the norm function produces. -{{{ switch to next slide, determinant }}} - Now let us find out the determinant of a the matrix m5. -The determinant of a square matrix can be obtained using the function +The determinant of a square matrix can be obtained by using the function ``det()`` and the determinant of m5 can be found out as, :: det(m5) +Hence we get the determinant. + {{{ switch to next slide, eigen vectors and eigen values }}} The eigen values and eigen vector of a square matrix can be computed using the function ``eig()`` and ``eigvals()``. Let us find out the eigen values and eigen vectors of the matrix -m5. We can do it as, +m5. We find them as, :: eig(m5) - -.. #[punch: has the tuple word been introduced?] - Note that it returned a tuple of two matrices. The first element in the tuple are the eigen values and the second element in the tuple are -the eigen vectors. Thus the eigen values are, +the eigen vectors. Thus the eigen values are given by, :: eig(m5)[0] -and the eigen vectors are, +and the eigen vectors are given by, :: eig(m5)[1] @@ -279,15 +271,15 @@ The eigen values can also be computed using the function ``eigvals()`` as, Now let us learn how to do the singular value decomposition or S V D of a matrix. -Suppose M is an m×n matrix whose entries come from the field K, which +Suppose M is an m×n matrix, whose entries come from the field K, which is either the field of real numbers or the field of complex numbers. Then there exists a factorization of the form M = U\Sigma V star where U is an (m by m) unitary matrix over K, the matrix \Sigma is an -(m by n) diagonal matrix with nonnegative real numbers on the -diagonal, and V*, an (n by n) unitary matrix over K, denotes the +(m by n) diagonal matrix with non-negative real numbers on the +diagonal, and V* is an (n by n) unitary matrix over K,which denotes the conjugate transpose of V. Such a factorization is called the singular-value decomposition of M. @@ -299,21 +291,62 @@ The SVD of matrix m5 can be found as Notice that it returned a tuple of 3 elements. The first one U the next one Sigma and the third one V star. -{{{ switch to next slide, recap slide }}} +{{{ switch to summary slide }}} -So this brings us to the end of this tutorial. In this tutorial, we -learned about matrices, creating matrices, matrix operations, inverse -of matrices, determinant, norm, eigen values and vectors and singular -value decomposition of matrices. +This brings us to the end of the end of this tutorial.In this tutorial, +we have learnt to, -{{{ switch to next slide, thank you }}} + 1. Create matrices using arrays. + #. Add,subtract and multiply the elements of matrix. + #. Find out the inverse of a matrix,using the function ``inv()``. + #. Use the function ``det()`` to find the determinant of a matrix. + #. Calculate the norm of a matrix using the for loop and also using + the function ``norm()``. + #. Find out the eigen vectors and eigen values of a matrix, using + functions ``eig()`` and ``eigvals()``. + #. Calculate singular value decomposition(SVD) of a matrix using the + function ``svd()``. +{{{Show self assessment questions slide}}} + +Here are some self assessment questions for you to solve + +1. A and B are two array objects. Element wise multiplication in + matrices are done by, + + - A * B + - ``multiply(A, B)`` + - ``dot(A, B)`` + - ``element_multiply(A,B)`` + +2. ``eig(A)[1]`` and ``eigvals(A)`` are the same. + + - True + - False + +3. ``norm(A,ord='fro')`` is the same as ``norm(A)`` ? + + - True + - False + +{{{solution of self assessment questions on slide}}} + +And the answers, + +1. Element wise multiplication between two matrices, A and B is done as, + A * B + +2. False. + ``eig(A)[0]`` and ``eigvals(A)`` are same, that is both will give the + eigen values of matrrix A. + +3. ``norm(A,ord='fro')`` and ``norm(A)`` are same, since the order='fro' + stands for frobenius norm. Hence true. + + +{{{ switch to Thank you slide }}} + +Hope you have enjoyed this tutorial and found it useful. Thank you! -.. - Local Variables: - mode: rst - indent-tabs-mode: nil - sentence-end-double-space: nil - fill-column: 70 - End: + |