Python/C3/Matrices/English-timed - Revision history

PoojaMoolya at 11:41, 20 February 2017

2017-02-20T11:41:45Z

Gaurav at 09:18, 10 July 2014

2014-07-10T09:18:47Z

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Sneha at 06:19, 20 March 2013

2013-03-20T06:19:50Z

Sneha at 11:22, 18 March 2013

2013-03-18T11:22:21Z

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Minal: Created page with '{| border=1 !Visual Cue !Narration |- | 0:01 | Hello friends and welcome to the tutorial on 'Matrices'. |- | 0:05 | At the end of this tutorial, you will be able to, # Create …'

2013-01-02T11:05:37Z

Created page with '{| border=1 !Visual Cue !Narration |- | 0:01 | Hello friends and welcome to the tutorial on 'Matrices'. |- | 0:05 | At the end of this tutorial, you will be able to, # Create …'

New page

{| border=1
!Visual Cue
!Narration
|-
| 0:01
| Hello friends and welcome to the tutorial on 'Matrices'.

|-
| 0:05
| 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.

|-
| 0:31
| 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".

|-
| 0:42
| Let us start our ipython interpreter with pylab loaded

|-
|0:47
|Type ipython hypen pylab

|-
| 0:52
| All matrix operations are done using arrays.

|-
| 0:55
| Thus all the operations on arrays are valid on matrices also.

|-
| 1:00
| A matrix may be created as,

|-
| 1:02
|Type in the terminal m1 = array within bracket and square bracket 1 comma 2 comma 3 comma 4 hit enter.

|-
| 1:16
| Using the method <tt>shape</tt>, we can find out the shape or size of the matrix,

|-
|1:20
|Type m1 dot shape and hit enter

|-
|1:27
|We can see the output

|-
| 1:29
| Since it is a one row four column matrix it returned a tuple, one by four.

|-
| 1:46
| A list can also be converted to a matrix as follows,

|-
|1:50
|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

|-
|2:28
|Sorry you have to type l1 array .

|-
| 2:35
| Pause the video here, try out the following exercise and resume the video.

|-
| 2:43
| Create a two dimensional matrix m3 of order 2 by 4 with elements 5, 6, 7, 8, 9, 10, 11, 12.

|-
| 2:51
| Switch to terminal for solution.

|-
| 2:54
| m3 can be created as,

|-
|2:56
|Type m3 = array within closing bracket within square bracket square bracket 5 comma 6 comma 7 comma 8 comma in another square bracket 9 comma 10 comma 11 comma 12

|-
| 3:31
| Let us now move to matrix operations.

|-
| 3:34
| We can do matrix addition and subtraction easily.

|-
| 3:37
| m3+m2 does element by element addition, that is matrix addition.

|-
| 3:43
| Note that both the matrices should be of the same order.

|-
|3:47
|Type m3+m2 and hit enter so you can see the output.

|-
| 3:55
| Similarly,m3 minus m2 does matrix subtraction, that is element by element subtraction.

|-
|4:02
|You can try out by typing m3 minus m2

|-
| 4:09
| Now let us try,matrix multiplication

|-
|4:13
|Type m3 star m2

|-
| 4:20
| Note that in arrays <tt>m3 star m2</tt> does element wise multiplication and not matrix multiplication,

|-
| 4:28
| Matrix multiplication in matrices are done using the function <tt>dot()</tt>

|-
|4:37
|Type dot within bracket m3 comma m2 and hit enter.

|-
|4:47
|So we can see error value show in the command.

|-
| 4:50
| Due to size mismatch, the multiplication could not be done and it returned an error.

|-
| 4:56
| Now let us see an example for matrix multiplication.

|-
| 5:00
| 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.

|-
| 5:11
| Thus let us first create two matrices which are compatible for multiplication.

|-
|5:16
|Type m1.shape and hit enter

|-
| 5:24
| matrix m1 is of the shape one by four,

|-
| 5:28
| let us create another one, of the order four by two,

|-
|5:33
|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

|-
| 6:10
| Thus the <tt>dot()</tt> function is used for matrix multiplication.

|-
| 6:15
| As we already learnt in arrays, the function <tt>identity()</tt> which creates an identity matrix of the order n by n,

|-
| 6:24
| the function <tt>zeros()</tt> which creates a matrix of the order m by n with all zeros,

|-
| 6:30
| the function <tt>zeros like function()</tt> which creates a matrix with zeros with the shape of the matrix passed,

|-
| 6:39
| the function <tt>ones()</tt> which creates a matrix of order m by n with all ones,

|-
| 6:47
| the function <tt>ones underscore like()</tt> which creates a matrix with ones with the shape of the matrix passed;

|-
| 6:53
| all these functions can also be used with matrices.

|-
| 6:57
|So now Let us see, how to find out the transpose of a matrix we can do,

|-
|7:03
|Type print m4
m4 dot T

|-
| 7:14
| As you saw, Matrix name dot capital T will give the transpose of a matrix

|-
| 7:21
|Pause the video here, try out the following exercise and resume the video.

|-
| 7:26
| Find out the Frobenius norm of inverse of a 4 by 4 matrix, the matrix being,

|-
|7:33
|m5 = arange within bracket 1 comma 17 dot reshape within bracket 4 comma 4

|-
| 7:44
|The Frobenius norm of a matrix is defined as, the square root of the sum of the absolute squares of its elements

|-
| 7:54
| Switch to terminal for the solution

|-
| 7:58
| Let us create the matrix m5 by using the data provided in the question

|-
|8:03
|Type m5 = arange within bracket 1 comma 17.reshape within square bracket 4 comma 4
print m5

|-
| 8:20
| 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.

|-
| 8:33
| The Frobenius norm of a matrix is defined as square root of sum of squares of elements in the matrix.

|-
| 8:41
| The inverse of a matrix can be found using the function <tt>inv(A)</tt>.

|-
| 8:47
| Type in the terminal im5 = inv within bracket m5

|-
| 8:57
| And the Frobenius norm of the matrix <tt>im5</tt> can be found out as,
sum = 0

for each in im5 dot flatten function():
sum plus= each star each
print sqrt within bracket sum

|-
| 9:52
| Thus we have successfully obtained the Frobenius norm of the matrix m5

|-
| 9:58
| 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 = within 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 <tt>norm()</tt>.

|-
| 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 <tt>norm</tt> to read up more about <tt>ord</tt> 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 <tt>det()</tt> 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 <tt>eig()</tt> and <tt>eigvals()</tt>.

|-
| 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 within 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 <tt>eigvals()</tt> 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 = USigma 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 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.

|-
| 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 <tt>inv()</tt>.

|-
| 15:32
| 4. Use the function <tt>det()</tt> 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 <tt>norm()</tt>.

|-
| 15:43
| 6. Finding out the eigen vectors and eigen values of a matrix, using functions <tt>eig()</tt> and <tt>eigvals()</tt>.

|-
| 15:50
| 7. Calculate singular value decomposition(SVD) of a matrix using the function <tt>svd()</tt>.

|-
| 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
<tt>multiply within bracket A comma B</tt>
<tt>dot within bracket A comma B</tt>
<tt>element underscore multiply within bracket A comma B</tt>

|-
| 16:19
| 2. <tt><nowiki>eig within bracket A within square bracket 1</nowiki></tt> and <tt>eigvals within bracket A</tt> are the same.Is it True or False?

|-
| 16:31
| 3. <tt>norm within bracket A comma ord= within is equal to fro</tt> is the same as <tt>norm within bracket A</tt>. 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. <tt><nowiki>eig within bracket A within square bracket 0</nowiki></tt> and <tt>eigvals within bracket A</tt> are same, that is both will give the eigen values of matrix A.

|-
| 17:06
|3. <tt>norm within bracket A comma ord=is equal to fro</tt> and <tt>norm(A)</tt> 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!

|}

@@ Line 10: / Line 10: @@
 | 00:05
 | At the end of this tutorial, you will be able to,
 Create matrices using data.
-# Create matrices using data.
+Create matrices from lists.
-# Create matrices from lists.
+Do basic matrix operations like addition,multiplication.
-# 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.
-# 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.
 |-
@@ Line 63: / Line 62: @@
 |  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 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
- Type m2 = array within bracket 11
 |-
@@ Line 176: / Line 173: @@
 |-
 |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
+|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
- Type dot within bracket m1 comma  m4
 |-
@@ Line 213: / Line 209: @@
 |-
 |07:03
-|Type print m4
+|Type print m4,  m4 dot T
   m4 dot T
 |-
@@ Line 246: / Line 241: @@
 |-
 |08:03
-|Type m5 = arrange  within bracket 1 comma 17.reshape within square bracket 4 comma 4
+|Type m5 = arrange  within bracket 1 comma 17.reshape within square bracket 4 comma 4. Then type  print m5
 Then type  print m5
 |-
@@ Line 267: / Line 261: @@
 |-
 | 08:57
-| And the Frobenius norm of the matrix  im5  can be found out as,
+| And the Frobenius norm of the matrix  im5  can be found out as, sum = 0
- sum = 0
-−
 for each in im5 dot flatten function():
-    sum plus= each star each
+sum plus= each star each
 print sqrt  within bracket sum
@@ Line 298: / Line 290: @@
 |sum underscore rows =  square bracket
 for i in im5 colon
-    sum underscore rows.append  within bracket abs  within bracket i.sum()
+sum underscore rows.append  within bracket abs  within bracket i.sum()
 print max within square bracket sum underscore rows
@@ Line 433: / Line 425: @@
 |-
 | 15:22
-| In this tutorial, we have learnt to, 1. Create matrices using arrays.
+| In this tutorial, we have learnt to, Create matrices using arrays.
 |-
 |  15:25
-| 2. Add,subtract and multiply the elements of matrix.
+| Add,subtract and multiply the elements of matrix.
 |-
 |  15:28
-| 3. Find out the inverse of a matrix,using the function  inv() .
+|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.
+| 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()
+| 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()
+|  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() .
+| Calculate singular value decomposition(SVD) of a matrix using the function svd() .
 |-
@@ Line 464: / Line 456: @@
 |-
 |  16:01
-| 1. A and B are two array objects. Element wise multiplication in matrices are done by,
+| A and B are two array objects. Element wise multiplication in matrices are done by,
-    A * B
+A * B
-   multiply within bracket A comma  B
+multiply within bracket A comma  B
-   dot within bracket A comma  B
+dot within bracket A comma  B
-   element underscore multiply 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?
+| 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?
+|norm  within bracket A comma ord= within is equal to fro  is the same as  norm within bracket A . Is it True or False?
 |-
@@ Line 484: / Line 476: @@
 |-
 |  16:47
-| 1. Element wise multiplication between two matrices, A and B is done as, A into B
+| 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.
+|  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.
+|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.
 |-

@@ Line 21: / Line 21: @@
 |-
 |  0:42
-| Let us start our ipython interpreter with pylab loaded
+| Let us start our ipython interpreter by pylab loaded
 |-
 |0:47
-|Type ipython hypen pylab
+|Type ipython hypen pylab in terminal
 |-
@@ Line 33: / Line 33: @@
 |-
 |  0:55
-| Thus all the operations on arrays are valid on matrices also.
+| Thus all the operations on arrays are valid on matrices only.
 |-
@@ Line 91: / Line 91: @@
 |-
 |2:56
-|Type m3 = array within closing bracket within square bracket square bracket 5 comma 6 comma 7 comma 8 comma in another square bracket 9 comma 10 comma 11 comma 12
+|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
 |-
@@ Line 229: / Line 229: @@
 |-
 |7:33
-|m5 = arrange  within bracket 1 comma 17 dot reshape  within bracket 4 comma 4
+|m5 = arrange  within bracket 1 comma 17 dot reshape to 4 comma 4
 |-
@@ Line 246: / Line 246: @@
 |8:03
 |Type m5 = arrange  within bracket 1 comma 17.reshape within square bracket 4 comma 4
-  print m5
+Then type  print m5
 |-
@@ Line 295: / Line 295: @@
 |-
 |10:20
-|sum underscore rows = within square bracket
+|sum underscore rows =  square bracket
 for i in im5 colon
      sum underscore rows.append  within bracket abs  within bracket i.sum()
@@ Line 374: / Line 374: @@
 |-
 |  13:11
-| Thus eigen values are given by,eig  within bracket m5  within square bracket 0
+| Thus eigen values are given by,eig  within bracket m5  and in square bracket 0
 |-
@@ Line 382: / Line 382: @@
 |-
 | 13:44
-| The eigen values can also be computed using the function <tt>eigvals()</tt> as,
+| The eigen values can also be computed using the function eigvals() as,
 |-
@@ Line 400: / Line 400: @@
 | Then there exists a factorization of the form
-M = USigma V star
+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 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.
+| 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.
 |-