Python/C3/Matrices/English-timed

Time	Narration
00:01	Hello friends and welcome to the tutorial on 'Matrices'.
00:05	At the end of this tutorial, you will be able to, 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.
00:31	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".
00:42	Let us start our ipython interpreter by pylab loaded
00:47	Type ipython hypen pylab in terminal
00:52	All matrix operations are done using arrays.
00:55	Thus all the operations on arrays are valid on matrices only.
01:00	A matrix may be created as,
01:02	Type in the terminal m1 = array within bracket and square bracket 1 comma 2 comma 3 comma 4 hit enter.
01:16	Using the method shape , we can find out the shape or size of the matrix,
01:20	Type m1 dot shape and hit enter
01:27	We can see the output
01:29	Since it is a one row four column matrix it returned a tuple, one by four.
01:46	A list can also be converted to a matrix as follows,
01:50	Type l1 = within square bracket square bracket 1 comma 2 comma 3 comma 4 comma in another square bracket 5 comma 6 comma 7 comma 8 Type m2 = array within bracket 11
02:28	Sorry you have to type l1 array .
02:35	Pause the video here, try out the following exercise and resume the video.
02:43	Create a two dimensional matrix m3 of order 2 by 4 with elements 5, 6, 7, 8, 9, 10, 11, 12.
02:51	Switch to terminal for solution.
02:54	m3 can be created as,
02:56	Type m3 = array within closing bracket inside square bracket square bracket 5 comma 6 comma 7 comma 8 comma in another square bracket 9 comma 10 comma 11 comma 12
03:31	Let us now move to matrix operations.
03:34	We can do matrix addition and subtraction easily.
03:37	m3+m2 does element by element addition, that is matrix addition.
03:43	Note that both the matrices should be of the same order.
03:47	Type m3+m2 and hit enter so you can see the output.
03:55	Similarly,m3 minus m2 does matrix subtraction, that is element by element subtraction.
04:02	You can try out by typing m3 minus m2
04:09	Now let us try,matrix multiplication
04:13	Type m3 star m2
04:20	Note that in arrays m3 star m2 does element wise multiplication and not matrix multiplication,
04:28	Matrix multiplication in matrices are done using the function dot()
04:37	Type dot within bracket m3 comma m2 and hit enter.
04:47	So we can see error value show in the command.
04:50	Due to size mismatch, the multiplication could not be done and it returned an error.
04:56	Now let us see an example for matrix multiplication.
05:00	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.
05:11	Thus let us first create two matrices which are compatible for multiplication.
05:16	Type m1.shape and hit enter
05:24	matrix m1 is of the shape one by four,
05:28	let us create another one, of the order four by two,
05:33	So type m4 = array in closing bracket within square bracket within square bracket 1 comma 2 comma within square bracket 3 comma 4 comma within square bracket 5 comma 6 within square bracket comma within square bracket 7 comma 8 Type dot within bracket m1 comma m4
06:10	Thus the dot() function is used for matrix multiplication.
06:15	As we already learnt in arrays, the function identity() which creates an identity matrix of the order n by n,
06:24	the function zeros() which creates a matrix of the order m by n with all zeros,
06:30	the function zeros like function() which creates a matrix with zeros with the shape of the matrix passed,
06:39	the function ones() which creates a matrix of order m by n with all ones,
06:47	the function ones underscore like() which creates a matrix with ones with the shape of the matrix passed;
06:53	all these functions can also be used with matrices.
06:57	So now Let us see, how to find out the transpose of a matrix we can do,
07:03	Type print m4 m4 dot T
07:14	As you saw, Matrix name dot capital T will give the transpose of a matrix
07:21	Pause the video here, try out the following exercise and resume the video.
07:26	Find out the Frobenius norm of inverse of a 4 by 4 matrix, the matrix being,
07:33	m5 = arrange within bracket 1 comma 17 dot reshape to 4 comma 4
07:44	The Frobenius norm of a matrix is defined as, the square root of the sum of the absolute squares of its elements
07:54	Switch to terminal for the solution
07:58	Let us create the matrix m5 by using the data provided in the question
08:03	Type m5 = arrange within bracket 1 comma 17.reshape within square bracket 4 comma 4 Then type print m5
08:20	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.
08:33	The Frobenius norm of a matrix is defined as square root of sum of squares of elements in the matrix.
08:41	The inverse of a matrix can be found using the function inv(A) .
08:47	Type in the terminal im5 = inv within bracket m5
08:57	And the Frobenius norm of the matrix im5 can be found out as, sum = 0 for each in im5 dot flatten function(): sum plus= each star each print sqrt within bracket sum
09:52	Thus we have successfully obtained the Frobenius norm of the matrix m5
09:58	Pause the video here, try out the following exercise and resume the video.
10:04	Find out the infinity norm of the matrix im5.
10:08	The infinity norm of a matrix is defined as the maximum value of sum of the absolute of elements in each row.
10:16	Switch to terminal for the solution
10:20	sum underscore rows = square bracket for i in im5 colon sum underscore rows.append within bracket abs within bracket i.sum() print max within square bracket sum underscore rows
11:01	Well! to find out the Frobenius norm and Infinity norm we have an even easier method, and let us see that now.
11:10	The norm of a matrix can be found out using the method norm().
11:19	In order to find out the Frobenius norm of the matrix im5, we do,
11:25	In the terminal type norm within bracket im5 and hit enter
11:34	And to find out the Infinity norm of the matrix im5, we do,
11:39	norm within bracket im5,ord=inf
11:51	This is easier when compared to the code we wrote.
11:55	Read the documentation of norm to read up more about ord and the possible type of norms the norm function produces.
12:04	Now let us find out the determinant of a the matrix m5.
12:11	The determinant of a square matrix can be obtained by using the function det() and the determinant of m5 can be found out as,
12:20	So type det within bracket m5
12:26	Hence we get the determinant.
12:29	Let us now move on to eigen vectors and eigen values
12:34	The eigen values and eigen vector of a square matrix can be computed using the function eig() and eigvals().
12:46	Let us find out the eigen values and eigen vectors of the matrix m5. We find them as,
12:53	Typing eig within bracket m5 in the terminal.
13:02	Note that it returned a tuple of two matrices.
13:06	The first element in the tuple are the eigen values and the second element in the tuple are the eigen vectors.
13:11	Thus eigen values are given by,eig within bracket m5 and in square bracket 0
13:30	and the eigen vectors are given by,eig within bracket m5 within square bracket 1
13:44	The eigen values can also be computed using the function eigvals() as,
13:50	Typing on the terminal eigvals within bracket m5
13:58	Now let us learn how to do the singular value decomposition or S V D of a matrix.
14:06	Suppose M is an m (cross) in matrix, whose entries come from the field K, which is either the field of real numbers or the field of complex numbers.
14:18	Then there exists a factorization of the form M = U Sigma V star
14:25	where U is an (m by m) unitary matrix over K, the matrix Sigma is an (m by n) diagonal matrix and the 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.
14:53	Such a factorization is called the singular-value decomposition of M.
14:58	The SVD of matrix m5 can be found as
15:01	So now open the terminal and type svd within brackets m5
15:09	Notice that it returned a tuple of 3 elements.
15:12	The first one U the next one Sigma and the third one V star
15:19	This brings us to the end of the end of this tutorial.
15:22	In this tutorial, we have learnt to, 1. Create matrices using arrays.
15:25	2. Add,subtract and multiply the elements of matrix.
15:28	3. Find out the inverse of a matrix,using the function inv() .
15:32	4. Use the function det() to find the determinant of a matrix.
15:36	5. Calculate the norm of a matrix using the for loop and also using the function norm()
15:43	6. Finding out the eigen vectors and eigen values of a matrix, using functions eig() and eigvals()
15:50	7. Calculate singular value decomposition(SVD) of a matrix using the function svd() .
15:58	Here are some self assessment questions for you to solve
16:01	1. A and B are two array objects. Element wise multiplication in matrices are done by, A * B multiply within bracket A comma B dot within bracket A comma B element underscore multiply within bracket A comma B
16:19	2. eig within bracket A within square bracket 1 and eigvals within bracket A are the same.Is it True or False?
16:31	3. norm within bracket A comma ord= within is equal to fro is the same as norm within bracket A . Is it True or False?
16:43	Look at the answers,
16:47	1. Element wise multiplication between two matrices, A and B is done as, A into B
16:53	2. False. eig within bracket A within square bracket 0and eigvals within bracket A are same, that is both will give the eigen values of matrix A.
17:06	3. norm within bracket A comma ord=is equal to fro and norm(A) are same, since the order=is equal to fro stands for Frobenius norm.
17:22	Hence answer is true.
17:26	Hope you have enjoyed this tutorial and found it useful.
17:30	Thank you!

Contributors and Content Editors

Gaurav, Minal, PoojaMoolya, Sneha